Chapter 5 - The Riemann Integral, Measuring Areas Under Curves, Properties and Classes of Integrable Functions, Properties of Integrals, Linearity, Monotonicity, and the Mean Value Theorem for Integrals, Fundamental Theorem and Applications

5.1 Definition and Integrability Criteria: Slicing and Summing Areas

We now turn to integration, the other fundamental operation in calculus. While differentiation is about rates of change and slopes, integration is about accumulation and areas. The Riemann Integral, named after Bernhard Riemann, provides a rigorous way to define the “area under a curve”.

Imagine you have a curve defined by a function $f(x)$ on an interval $[a,b]$. We want to find the area between this curve, the x-axis, and the vertical lines $x=a$ and $x=b$.

The basic idea of the Riemann Integral is to approximate this area by dividing the interval $[a,b]$ into many small subintervals, constructing rectangles on each subinterval whose height is given by the function value, and then summing up the areas of these rectangles. As we make the subintervals smaller and smaller, this approximation should get closer and closer to the “true” area under the curve.

To make this precise, we need the concept of a partition.

Definition 5.1.1: Partition of an Interval

A partition $P$ of a closed interval $I=[a,b]$ is a finite set of points $\{x_0,x_1,\dots ,x_n\}$ such that $a=x_0<x_1<\cdots <x_n=b$. The partition divides the interval $[a,b]$ into $n$ subintervals $[x_{i-1},x_i]$, for $i=1,2,\dots ,n$.

Think of a partition as slicing the interval $[a,b]$ into smaller pieces.

For each subinterval $[x_{i-1},x_i]$, let’s consider a rectangle. We need to decide on the height of this rectangle. For the lower sum, we choose the minimum value of $f(x)$ on the subinterval as the height. For the upper sum, we choose the maximum value.

Let:

• $f_i=\inf \{f(x):x\in [x_{i-1},x_i]\}$ be the infimum (greatest lower bound) of $f(x)$ on the $i$-th subinterval.
• $F_i=\sup \{f(x):x\in [x_{i-1},x_i]\}$ be the supremum (least upper bound) of $f(x)$ on the $i$-th subinterval.
• ${\delta }_i=x_i-x_{i-1}$ be the width of the $i$-th subinterval.

Then we define:

• Lower Sum: $s(f,P)={\sum }_{i=1}^n{f}_i{\delta }_i$ (sum of areas of “inner” rectangles, always under the curve)
• Upper Sum: $S(f,P)={\sum }_{i=1}^n{F}_i{\delta }_i$ (sum of areas of “outer” rectangles, always over or touching the curve)

If $f(x)\ge 0$, the lower sum is an underestimate of the area, and the upper sum is an overestimate.

For any bounded function $f$ on $[a,b]$, and any partition $P$, we will always have:

Now, we consider what happens as we refine the partition, making the subintervals smaller and smaller. A refinement $P^{\prime }$ of a partition $P$ is a new partition that includes all the points of $P$ and possibly more. Refining a partition means adding more slices, making the rectangles narrower.

Lemma 5.1.2 (1): Refinement Improves Approximation

If $P^{\prime }$ is a refinement of $P$, then: $s(f,P)\le s(f,P^{\prime })\le S(f,P^{\prime })\le S(f,P)$

Refining a partition never makes the lower sum smaller or the upper sum larger. It generally brings the lower and upper sums closer together, giving a better approximation of the area.

For any two partitions $P_1$ and $P_2$, even if one is not a refinement of the other, the lower sum of $P_1$ is always less than or equal to the upper sum of $P_2$.

Lemma 5.1.2 (2): Lower Sums Always Below Upper Sums

For any partitions $P_1$ and $P_2$, $s(f,P_1)\le S(f,P_2)$.

This is a crucial observation. The set of all lower sums is always bounded above by any upper sum, and the set of all upper sums is always bounded below by any lower sum.

This leads us to define the Darboux integral sums:

• Lower Darboux Integral: $s(f)={\sup }_{P\in \mathcal{P}(I)}s(f,P)$ (the supremum of all lower sums)
• Upper Darboux Integral: $S(f)={\inf }_{P\in \mathcal{P}(I)}S(f,P)$ (the infimum of all upper sums)

Since lower sums are always less than or equal to upper sums, we always have $s(f)\le S(f)$.

Definition 5.1.3: Riemann Integrability

A bounded function $f:[a,b]\to \mathbb{R}$ is Riemann integrable (or simply integrable) if the lower Darboux integral equals the upper Darboux integral:

$s(f)=S(f)$

In this case, the common value is called the Riemann integral of $f$ from $a$ to $b$, denoted by ${\int }_a^{b}f(x)dx=s(f)=S(f)$.

If a function is Riemann integrable, there is a unique “true” area under its curve, which is captured by both the supremum of lower sums and the infimum of upper sums.

Riemann Integrability Criterion:

How do we determine if a function is Riemann integrable? The following theorem gives a practical criterion:

Theorem 5.1.4: Riemann Integrability Criterion

A bounded function $f:[a,b]\to \mathbb{R}$ is Riemann integrable if and only if for every $ϵ>0$, there exists a partition $P$ of $[a,b]$ such that $S(f,P)-s(f,P)<ϵ$.

This criterion says that a function is integrable if we can make the difference between upper and lower sums arbitrarily small by choosing a fine enough partition. This means the “inner” and “outer” rectangular approximations can be made to agree to within any desired accuracy.

Examples of Integrable and Non-Integrable Functions:

• Example 1: $f(x)=x$ is integrable. We showed that for $f(x)=x$, the limit of upper and lower sums is the same, so it is integrable.
• Example 2: Dirichlet Function is not integrable. The Dirichlet function is 1 on rationals and 0 on irrationals. For any interval, the supremum is 1 and the infimum is 0. Thus, upper sums are always 1, lower sums are always 0, and the difference is always 1, no matter how fine the partition. So, the Dirichlet function is not Riemann integrable.
• Example 3: Thomae’s Function (Modified Dirichlet) is integrable. This function is 0 for irrationals and $1/q$ for rationals $p/q$. It is discontinuous at every rational, but continuous at every irrational. Despite its many discontinuities, it is Riemann integrable, and its integral is 0. This example shows that Riemann integrability is more general than continuity.

In the next section, we’ll explore important classes of functions that are guaranteed to be Riemann integrable.

5.2 Integrable Functions: Which Functions Can We Integrate?

Not all bounded functions are Riemann integrable (as the Dirichlet function showed). But many important classes of functions are integrable.

Theorem 5.2.1: Properties of Riemann Integrable Functions

Let $f,g:[a,b]\to \mathbb{R}$ be bounded and Riemann integrable functions, and let $\lambda \in \mathbb{R}$. Then the following functions are also Riemann integrable on $[a,b]$:

1. Sum: $f+g$
2. Scalar Multiple: $\lambda \cdot f$
3. Product: $f\cdot g$
4. Absolute Value: $|f|$
5. Maximum and Minimum: $\max (f,g)$, $\min (f,g)$
6. Quotient: $f/g$, if there exists $\beta >0$ such that $|g(x)|\ge \beta$ for all $x\in [a,b]$ (i.e., $g$ is bounded away from zero).

These properties tell us that the set of Riemann integrable functions is “closed” under basic algebraic operations. If you start with integrable functions, you can combine them in these ways and still get integrable functions (with some conditions for division).

Proof of Integrability Properties (Sketch - Sum and Product)

We use the Riemann Integrability Criterion (Theorem 5.1.4): $f$ is integrable iff for every $ϵ>0$, there exists a partition $P$ with $S(f,P)-s(f,P)<ϵ$.

1. Sum $(f+g)$: Use the fact that the difference between max and min of $(f+g)$ on an interval is bounded by the sum of differences for $f$ and $g$ separately (Remark 5.2.2 in the script). Given $ϵ>0$, choose partitions $P_1,P_2$ such that $S(f,P_1)-s(f,P_1)<ϵ$ and $S(g,P_2)-s(g,P_2)<ϵ$. Let $P=P_1\cup P_2$ be their common refinement. Then $S(f+g,P)-s(f+g,P)\le [S(f,P)-s(f,P)]+[S(g,P)-s(g,P)]<2ϵ$.

2. Product $(f\cdot g)$: Use the fact that the difference between max and min of $(f\cdot g)$ is bounded by an expression involving bounds on $f$ and $g$ and the differences for $f$ and $g$ separately. Similar argument using common refinement to show integrability of $f\cdot g$.

Continuous Functions are Integrable: Smoothness Guarantees Area Measurement

A very important class of integrable functions is the set of continuous functions. If a function is continuous on a closed interval, it is guaranteed to be Riemann integrable. This is a cornerstone result of integral calculus.

Theorem 5.2.7: Continuity Implies Integrability

If $f:[a,b]\to \mathbb{R}$ is continuous on $[a,b]$, then $f$ is Riemann integrable on $[a,b]$.

Proof of Continuity Implies Integrability (Using Uniform Continuity)

1. Uniform Continuity: Since $f$ is continuous on the compact interval $[a,b]$, it is uniformly continuous (Theorem 5.2.6 in script, Heine-Cantor Theorem). This means for every $ϵ>0$, there exists a $\delta >0$ such that for any $x,y\in [a,b]$, if $|x-y|<\delta$, then $|f(x)-f(y)|<ϵ$. The $\delta$ depends only on $ϵ$, not on the point $x$.
2. Choose a partition with small subintervals: Given $ϵ>0$, choose $\delta >0$ from uniform continuity. Choose a partition $P$ such that the width of each subinterval ${\delta }_i=x_i-x_{i-1}<\delta$.
3. Bound the difference between upper and lower sums: For each subinterval $[x_{i-1},x_i]$, since the width is less than $\delta$, the difference between the maximum value $F_i$ and minimum value $f_i$ of $f$ on this subinterval is less than $ϵ$: $F_i-f_i\le ϵ$.
4. Show $S(f,P)-s(f,P)<ϵ(b-a)$: $S(f,P)-s(f,P)={\sum }_{i=1}^n(F_i-f_i){\delta }_i\le {\sum }_{i=1}^nϵ{\delta }_i=ϵ{\sum }_{i=1}^n{\delta }_i=ϵ(b-a)$.
5. Apply Riemann Integrability Criterion: For any ${ϵ}^{\prime }>0$, choose $ϵ=\frac{{ϵ}^{\prime }}{b-a}$. Then we found a partition $P$ with $S(f,P)-s(f,P)<{ϵ}^{\prime }$. By Theorem 5.1.4, $f$ is Riemann integrable.

Monotone Functions are Integrable:

Another important class of integrable functions is the set of monotone functions (functions that are always increasing or always decreasing).

Theorem 5.2.8: Monotonicity Implies Integrability

If $f:[a,b]\to \mathbb{R}$ is a monotone function on $[a,b]$, then $f$ is Riemann integrable on $[a,b]$.

Monotone functions can have discontinuities (jumps), but they are still “well-behaved” enough to be Riemann integrable.

These theorems significantly expand the class of functions we know are Riemann integrable, including continuous functions, polynomials, rational functions (where defined), exponential functions, trigonometric functions, absolute values, max/min of integrable functions, and monotone functions. This gives us a rich toolkit for working with integrals.

5.3 Properties of the Riemann Integral: Linearity and Monotonicity - Making Integrals Easier to Work With

The Riemann integral is not just a definition; it’s a powerful tool with useful properties that make it easier to work with and apply. Two key properties are linearity and monotonicity.

Linearity of the Integral:

The integral is a linear operator. This means it respects addition and scalar multiplication.

Theorem 5.2.10: Linearity of the Integral

Let $f_1,f_2:[a,b]\to \mathbb{R}$ be Riemann integrable functions, and let ${\lambda }_1,{\lambda }_2\in \mathbb{R}$. Then the linear combination $({\lambda }_1{f}_1+{\lambda }_2{f}_2)$ is also Riemann integrable on $[a,b]$, and:

${\int }_a^b({\lambda }_1{f}_1(x)+{\lambda }_2{f}_2(x))dx={\lambda }_1{\int }_a^b{f}_1(x)dx+{\lambda }_2{\int }_a^b{f}_2(x)dx$

This is incredibly useful. It allows us to break down integrals of sums into sums of integrals, and pull out constant factors.

Proof of Linearity of the Integral (Using Riemann Sums)

Use Corollary 5.1.9: The integral $A={\int }_a^{b}f(x)dx$ is the limit of Riemann sums ${\sum }_{i=1}^{n}f({\xi }_i){\delta }_i$.

Show that the Riemann sum for $({\lambda }_1{f}_1+{\lambda }_2{f}_2)$ is a linear combination of Riemann sums for $f_1$ and $f_2$. Then take the limit as the partition becomes finer.

Monotonicity of the Integral:

The integral preserves order. If one function is always less than or equal to another, its integral will also be less than or equal.

Theorem 5.3.1: Monotonicity of the Integral

Let $f,g:[a,b]\to \mathbb{R}$ be Riemann integrable functions. If $f(x)\le g(x)$ for all $x\in [a,b]$, then:

${\int }_a^{b}f(x)dx\le {\int }_a^{b}g(x)dx$

Proof of Monotonicity of the Integral

Consider the difference $h(x)=g(x)-f(x)\ge 0$. By linearity, ${\int }_a^b(g(x)-f(x))dx={\int }_a^{b}g(x)dx-{\int }_a^{b}f(x)dx$. We need to show ${\int }_a^{b}h(x)dx\ge 0$.

Since $h(x)\ge 0$, for any partition $P$, the lower sum $s(h,P)={\sum }_{i=1}^n{\inf }_{x\in [x_{i-1},x_i]}h(x)\cdot {\delta }_i\ge 0$ (since $\inf h(x)\ge 0$).

Since the lower integral $s(h)={\sup }_{P}s(h,P)$, and all lower sums are non-negative, the supremum must also be non-negative: $s(h)={\int }_a^{b}h(x)dx\ge 0$.

Bounding the Absolute Value of an Integral:

Combining linearity and monotonicity, we get a useful bound on the absolute value of an integral.

Corollary 5.3.2: Bound on Absolute Value of Integral

If $f:[a,b]\to \mathbb{R}$ is Riemann integrable, then:

$\left|{\int }_a^{b}f(x)dx\right|\le {\int }_a^b|f(x)|dx$

Proof of Bound on Absolute Value

We use the inequalities $-|f(x)|\le f(x)\le |f(x)|$. By monotonicity of the integral, integrating these inequalities from $a$ to $b$ gives: $-{\int }_a^b|f(x)|dx\le {\int }_a^{b}f(x)dx\le {\int }_a^b|f(x)|dx$

This is equivalent to $\left|{\int }_a^{b}f(x)dx\right|\le {\int }_a^b|f(x)|dx$.

Cauchy-Schwarz Inequality for Integrals:

A powerful inequality relating integrals of products to integrals of squares:

Theorem 5.3.3: Cauchy-Schwarz Inequality for Integrals

Let $f,g:[a,b]\to \mathbb{R}$ be Riemann integrable functions. Then:

${\left({\int }_a^{b}f(x)g(x)dx\right)}^2\le \left({\int }_a^b[f(x){]}^{2}dx\right)\left({\int }_a^b[g(x){]}^{2}dx\right)$

Proof of Cauchy-Schwarz Inequality

Consider the integral of the non-negative function $(f(x)-tg(x){)}^2\ge 0$ for any real number $t$: $0\le {\int }_a^b(f(x)-tg(x){)}^{2}dx={\int }_a^b[f(x){]}^{2}dx-2t{\int }_a^{b}f(x)g(x)dx+t^2{\int }_a^b[g(x){]}^{2}dx$

Let $A={\int }_a^b[f(x){]}^{2}dx,B={\int }_a^{b}f(x)g(x)dx,C={\int }_a^b[g(x){]}^{2}dx$. Then $A-2tB+C{t}^2\ge 0$ for all $t\in \mathbb{R}$.

If $C=0$, then $B=0$ and the inequality holds trivially. If $C\ne 0$, the quadratic $C{t}^2-2Bt+A$ is always non-negative, so its discriminant must be non-positive: $(-2B{)}^2-4AC\le 0$, which simplifies to $B^2\le AC$, or ${\left({\int }_a^{b}f(x)g(x)dx\right)}^2\le \left({\int }_a^b[f(x){]}^{2}dx\right)\left({\int }_a^b[g(x){]}^{2}dx\right)$.

Mean Value Theorem for Integrals:

Analogous to the Mean Value Theorem for derivatives, there’s also a Mean Value Theorem for integrals.

Theorem 5.3.4: Mean Value Theorem for Integrals (Cauchy, 1821)

Let $f:[a,b]\to \mathbb{R}$ be a continuous function. Then there exists a point $\xi \in [a,b]$ such that:

${\int }_a^{b}f(x)dx=f(\xi )(b-a)$

This theorem says that the integral of a continuous function over an interval is equal to the value of the function at some point in the interval, multiplied by the length of the interval. The average value of $f$ over $[a,b]$ is attained by the function at some point $\xi$.

Proof of Mean Value Theorem for Integrals

By the Min-Max Theorem, $f$ attains a minimum value $f(u)$ and a maximum value $f(v)$ on $[a,b]$. Thus, for all $x\in [a,b]$, $f(u)\le f(x)\le f(v)$. By monotonicity of the integral, integrating these inequalities from $a$ to $b$ gives: ${\int }_a^{b}f(u)dx\le {\int }_a^{b}f(x)dx\le {\int }_a^{b}f(v)dx$ $f(u){\int }_a^{b}1dx\le {\int }_a^{b}f(x)dx\le f(v){\int }_a^{b}1dx$ $f(u)(b-a)\le {\int }_a^{b}f(x)dx\le f(v)(b-a)$

Dividing by $(b-a)$ (assuming $a<b$): $f(u)\le \frac{1}{b-a}{\int }_a^{b}f(x)dx\le f(v)$.

Let $C=\frac{1}{b-a}{\int }_a^{b}f(x)dx$. Then $f(u)\le C\le f(v)$. Since $f$ is continuous, by the Intermediate Value Theorem (IVT), there exists some point $\xi$ between $u$ and $v$ (and thus in $[a,b]$) such that $f(\xi )=C=\frac{1}{b-a}{\int }_a^{b}f(x)dx$. Multiplying by $(b-a)$ gives ${\int }_a^{b}f(x)dx=f(\xi )(b-a)$.

These properties – linearity, monotonicity, and the Mean Value Theorem – are fundamental tools for working with and applying integrals. They provide ways to simplify integrals, compare integrals, and relate integrals to function values, paving the way for the Fundamental Theorem of Calculus in the next section.

5.4 The Fundamental Theorem of Calculus: Derivatives and Integrals, Inverses of Each Other

The Fundamental Theorem of Calculus (FTC) is the cornerstone of calculus, linking differentiation and integration in a profound way. It essentially says that differentiation and integration are inverse operations of each other.

Let’s consider a continuous function $f:[a,b]\to \mathbb{R}$. We can define a new function $F(x)$ by integrating $f$ from a fixed point $a$ up to a variable point $x$:

The FTC tells us that this function $F(x)$ is not just any function; it’s actually an antiderivative of $f(x)$. Taking the derivative of $F(x)$ brings us back to the original function $f(x)$.

Theorem 5.4.1: Fundamental Theorem of Calculus (Part 1 - Differentiation of the Integral)

Let $f:[a,b]\to \mathbb{R}$ be a continuous function. Define the function $F:[a,b]\to \mathbb{R}$ by:

$F(x)={\int }_a^{x}f(t)dt,a\le x\le b$

Then $F$ is continuous on $[a,b]$, differentiable on $(a,b)$, and its derivative is equal to $f(x)$:

$F^{\prime }(x)=f(x)\text{for all }x\in [a,b]$

This is the first part of the FTC. It says that the integral of a continuous function, with a variable upper limit, is differentiable, and its derivative is the original function. Integration “undoes” differentiation in this sense.

Proof of Fundamental Theorem of Calculus (Part 1)

We want to show that ${\lim }_{x\to x_0}\frac{F(x)-F(x_0)}{x-x_0}=f(x_0)$.

Using the definition of $F(x)$: $F(x)-F(x_0)={\int }_a^{x}f(t)dt-{\int }_a^{x_0}f(t)dt={\int }_{x_0}^{x}f(t)dt$ (using property 5.2.9 of integrals).

By the Mean Value Theorem for Integrals (Theorem 5.3.4), there exists some $\xi$ between $x_0$ and $x$ such that ${\int }_{x_0}^{x}f(t)dt=f(\xi )(x-x_0)$.

Thus, $\frac{F(x)-F(x_0)}{x-x_0}=f(\xi )$. As $x\to x_0$, the point $\xi$ (which is between $x_0$ and $x$) must also approach $x_0$, so $\xi \to x_0$. Since $f$ is continuous, ${\lim }_{x\to x_0}f(\xi )=f(x_0)$.

Therefore, ${\lim }_{x\to x_0}\frac{F(x)-F(x_0)}{x-x_0}={\lim }_{x\to x_0}f(\xi )=f(x_0)$, which proves $F^{\prime }(x_0)=f(x_0)$.

Antiderivatives and Definite Integrals: The Other Direction

The second part of the FTC goes in the opposite direction. It tells us how to compute a definite integral (the integral with fixed limits $a$ and $b$) if we know an antiderivative of the function.

Definition 5.4.2: Antiderivative (Stammfunktion)

Let $f:[a,b]\to \mathbb{R}$ be a continuous function. A function $F:[a,b]\to \mathbb{R}$ is called an antiderivative (or primitive function or Stammfunktion) of $f$ if $F$ is differentiable on $[a,b]$ and $F^{\prime }(x)=f(x)$ for all $x\in [a,b]$.

Theorem 5.4.3: Fundamental Theorem of Calculus (Part 2 - Evaluation Theorem)

Let $f:[a,b]\to \mathbb{R}$ be a continuous function, and let $F$ be any antiderivative of $f$. Then:

${\int }_a^{b}f(x)dx=F(b)-F(a)$

This is the “evaluation theorem”. It provides a practical way to compute definite integrals. To find ${\int }_a^{b}f(x)dx$, you just need to find any antiderivative $F(x)$ of $f(x)$, and then evaluate $F(b)-F(a)$.

Proof of Fundamental Theorem of Calculus (Part 2)

From Part 1, we know that $G(x)={\int }_a^{x}f(t)dt$ is one antiderivative of $f$. Let $F$ be any antiderivative of $f$.

Since both $F$ and $G$ are antiderivatives of $f$, their derivatives are equal: $F^{\prime }(x)=f(x)$ and $G^{\prime }(x)=f(x)$, so $F^{\prime }(x)-G^{\prime }(x)=0$, or $(F-G{)}^{\prime }(x)=0$.

By Corollary 4.2.5 (1), if the derivative of a function is zero on an interval, the function must be constant. Thus, $F(x)-G(x)=C$ for some constant $C$. So $F(x)=G(x)+C$.

Now consider $F(b)-F(a)=[G(b)+C]-[G(a)+C]=G(b)-G(a)={\int }_a^{b}f(t)dt-{\int }_a^{a}f(t)dt={\int }_a^{b}f(t)dt$ (since ${\int }_a^{a}f(t)dt=0$).

Thus, ${\int }_a^{b}f(x)dx=F(b)-F(a)$ for any antiderivative $F$.

Examples of Using the FTC for Evaluation:

• Example 1: ${\int }_0^{a}xdx$. An antiderivative of $f(x)=x$ is $F(x)=\frac{x^2}{2}$. So, ${\int }_0^{a}xdx=F(a)-F(0)=\frac{a^2}{2}-\frac{0^2}{2}=\frac{a^2}{2}$.
• Example 2: ${\int }_0^a{x}^{2}dx$. An antiderivative of $f(x)=x^2$ is $F(x)=\frac{x^3}{3}$. So, ${\int }_0^a{x}^{2}dx=F(a)-F(0)=\frac{a^3}{3}-\frac{0^3}{3}=\frac{a^3}{3}$.

The FTC turns the problem of computing definite integrals into the problem of finding antiderivatives, which we can often do using differentiation rules in reverse and techniques like substitution and integration by parts.

5.5 Integration of Convergent Series: Interchanging Limits and Integrals

Can we integrate a convergent series term-by-term? If we have a series of functions ${\sum }_{n=0}^{\infty }{f}_n(x)$, and we know it converges to a function $f(x)$, is it true that the integral of the sum is the sum of the integrals?

In general, you cannot always interchange limits and integrals without extra conditions. However, if the convergence is uniform, then we can!

Theorem 5.5.1: Integration of Uniformly Convergent Series

Let $(f_n{)}_{n\ge 0}$ be a sequence of Riemann integrable functions on $[a,b]$ that converges uniformly to a function $f:[a,b]\to \mathbb{R}$. Then $f$ is Riemann integrable on $[a,b]$, and:

${\int }_a^b\left({\sum }_{n=0}^{\infty }{f}_n(x)\right)dx={\sum }_{n=0}^{\infty }\left({\int }_a^b{f}_n(x)dx\right)$

Uniform convergence is a stronger type of convergence than pointwise convergence. It ensures that the functions in the sequence approach the limit function “at the same rate” across the entire domain. Uniform convergence allows us to interchange limits and integrals, a very powerful tool in analysis.

Corollary 5.5.2: Term-by-Term Integration of Series

If the series ${\sum }_{n=0}^{\infty }{f}_n(x)$ converges uniformly on $[a,b]$, and each $f_n$ is Riemann integrable, then the sum function $f(x)={\sum }_{n=0}^{\infty }{f}_n(x)$ is Riemann integrable, and we can integrate term-by-term.

Example: Integrating Power Series

Power series, within their radius of convergence, converge uniformly on compact intervals. This means we can integrate them term-by-term.

Corollary 5.5.3: Term-by-Term Integration of Power Series

Let $f(x)={\sum }_{k=0}^{\infty }{c}_k{x}^k$ be a power series with positive radius of convergence $\rho >0$. Then for any $0\le r<\rho$, $f$ is integrable on $[-r,r]$, and for $x\in (-\rho ,\rho )$:

${\int }_0^{x}f(t)dt={\sum }_{n=0}^{\infty }\frac{c_n}{n+1}{x}^{n+1}$

Integrating a power series term-by-term just involves increasing the power of each term by 1 and dividing by the new power. The radius of convergence remains the same.

Example: Logarithm as Integral of Geometric Series

We can derive the power series for $\ln (1+x)$ by integrating the geometric series for $\frac{1}{1+x}$ term-by-term.

Example: Power Series for $\ln (1+x)$ by Integration

Start with the geometric series: $\frac{1}{1+x}=1-x+x^2-x^3+\dots$ for $|x|<1$. Integrate term-by-term from $0$ to $x$: ${\int }_0^x\frac{1}{1+t}dt={\int }_0^x(1-t+t^2-t^3+\dots )dt$ $\ln (1+x)-\ln (1+0)={\left[t-\frac{t^2}{2}+\frac{t^3}{3}-\frac{t^4}{4}+\dots \right]}_0^x$ $\ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots ={\sum }_{n=1}^{\infty }\frac{(-1{)}^{n-1}}{n}{x}^n$

This provides a powerful method for finding power series representations of functions by integrating known series.

5.6 Euler-Maclaurin Summation Formula: Approximating Sums with Integrals

The Euler-Maclaurin Summation Formula is a remarkable tool that connects sums and integrals. It provides a way to approximate sums by integrals, and also to approximate integrals by sums, with error terms expressed using Bernoulli numbers and higher derivatives. It’s a powerful technique for numerical analysis and asymptotic estimations.

This formula and the following sections (Stirling’s Formula, Improper Integrals, and Indefinite Integrals) delve into more advanced topics and applications of integration, showcasing the breadth and depth of integral calculus. We will not go into the details of these sections here, but they represent further directions and powerful tools within the realm of Riemann integration.

This concludes our journey through Chapter 5 and our exploration of the Riemann Integral. We’ve defined the integral rigorously, explored its properties, and seen how it connects to differentiation through the Fundamental Theorem of Calculus. We’ve also touched upon more advanced topics like integration of series and approximation formulas, hinting at the vast landscape of applications that integral calculus opens up.

5.7 Stirling’s Formula: Approximating Factorials for Large Numbers

The Stirling’s Formula provides a remarkable approximation for the factorial function $n!$ when $n$ is large. Factorials grow incredibly fast, and Stirling’s formula gives us a way to estimate their size using continuous functions and elementary operations.

Theorem 5.7.1: Stirling's Formula

As $n\to \infty$, $n!$ is asymptotically approximated by $\sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n$. More precisely:

${\lim }_{n\to \infty }\frac{n!}{\sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n}=1$

A more precise version is given by:

$n!=\sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n\cdot \exp \left(\frac{1}{12n}+R_3(n)\right)$

where $|R_3(n)|\le \frac{\sqrt{3}}{216}\cdot \frac{1}{n^2}$ for $n\ge 1$.

This formula is incredibly useful in various fields, including probability, statistics, and physics, where factorials appear frequently in calculations involving large numbers.

The derivation of Stirling’s formula in the script uses the Euler-Maclaurin Summation Formula (section 5.6), which provides a connection between sums and integrals. By applying the Euler-Maclaurin formula to the sum of logarithms, we can approximate $\ln (n!)=\ln 1+\ln 2+\cdots +\ln n$ using an integral and correction terms involving Bernoulli numbers and derivatives. Exponentiating the result then leads to Stirling’s formula.

5.8 Improper Integrals (Uneigentliche Integrale): Integrating to Infinity or Across Discontinuities

The Riemann integral, as we’ve defined it so far, applies to bounded functions on bounded intervals $[a,b]$. But what if we want to integrate over an unbounded interval like $[a,\infty )$ or if the function has a discontinuity within the interval? This leads us to improper integrals.

Improper Integrals of Type 1: Unbounded Intervals

For integration over an interval like $[a,\infty )$, we define the improper integral as a limit:

Definition 5.8.1: Improper Integral of Type 1

Let $f:[a,\infty )\to \mathbb{R}$ be bounded and Riemann integrable on $[a,b]$ for all $b>a$. If the limit

${\lim }_{b\to \infty }{\int }_a^{b}f(x)dx$

exists, we say that the improper integral ${\int }_a^{\infty }f(x)dx$ converges, and we define its value to be this limit:

${\int }_a^{\infty }f(x)dx={\lim }_{b\to \infty }{\int }_a^{b}f(x)dx$

If the limit does not exist, the improper integral diverges.

We essentially integrate over increasingly large bounded intervals $[a,b]$ and see if the integrals approach a finite limit as $b\to \infty$.

Examples:

• ${\int }_0^{\infty }{e}^{-x}dx=1$ (converges)
• ${\int }_1^{\infty }\frac{1}{x^{\alpha }}dx$ converges for $\alpha >1$ and diverges for $\alpha \le 1$.

Comparison Test for Improper Integrals:

Similar to series, we have comparison tests for improper integrals:

Lemma 5.8.3: Comparison Test for Improper Integrals

Let $f:[a,\infty )\to \mathbb{R}$ be bounded and integrable on $[a,b]$ for all $b>a$.

1. Convergence Test: If $|f(x)|\le g(x)$ for all $x\ge a$, and ${\int }_a^{\infty }g(x)dx$ converges, then ${\int }_a^{\infty }f(x)dx$ also converges.
2. Divergence Test: If $0\le g(x)\le f(x)$ for all $x\ge a$, and ${\int }_a^{\infty }g(x)dx$ diverges, then ${\int }_a^{\infty }f(x)dx$ also diverges.

McLaurin-Cauchy Integral Test:

A powerful test connecting series convergence to improper integral convergence:

Theorem 5.8.5: McLaurin-Cauchy Integral Test (1742)

Let $f:[1,\infty )\to [0,\infty )$ be a monotonically decreasing function. Then the series ${\sum }_{n=1}^{\infty }f(n)$ converges if and only if the improper integral ${\int }_1^{\infty }f(x)dx$ converges.

This test allows us to use integration to determine the convergence of certain types of series, and vice versa.

Improper Integrals of Type 2: Discontinuous Functions

For functions with a discontinuity at a point inside the interval of integration (or at an endpoint), we also define improper integrals using limits. For example, if $f$ is discontinuous at $x=a$, we define:

Definition 5.8.8: Improper Integral of Type 2 (Discontinuity at Endpoint)

Let $f:(a,b]\to \mathbb{R}$ be bounded and Riemann integrable on $[a+ϵ,b]$ for all $ϵ>0$. If the limit

${\lim }_{ϵ\to 0^{+}}{\int }_{a+ϵ}^{b}f(x)dx$

exists, we say that the improper integral ${\int }_a^{b}f(x)dx$ converges, and we define its value to be this limit:

${\int }_a^{b}f(x)dx={\lim }_{ϵ\to 0^{+}}{\int }_{a+ϵ}^{b}f(x)dx$

We approach the point of discontinuity by integrating over intervals that exclude the discontinuity, and see if the integrals approach a finite limit.

Example: ${\int }_0^1\frac{1}{x^{\alpha }}dx$ converges for $\alpha <1$ and diverges for $\alpha \ge 1$.

5.9 Indefinite Integral (Das unbestimmte Integral): The Family of Antiderivatives

The indefinite integral (unbestimmte Integral) is another name for the antiderivative. It represents the family of all functions whose derivative is the given function. We write:

where $F(x)$ is any antiderivative of $f(x)$, and $C$ is an arbitrary constant of integration. The indefinite integral is not a definite number, but a family of functions.

The script provides a table of basic indefinite integrals and demonstrates techniques like integration by parts and substitution for finding antiderivatives. These are essential tools for evaluating both indefinite and definite integrals.

This concludes our summary of Chapter 5 and the entire Analysis 1 script. We’ve covered the fundamental concepts of the Riemann integral, its properties, important classes of integrable functions, techniques of integration, improper integrals, and connections to series and approximation formulas. This provides a solid foundation in the basics of real analysis.

Previous Chapter: Chapter 4 - Differentiable Functions, The Slope of a Curve, Rules for Differentiation, Mean Value Theorem and Beyond