Lecture 33: Applications of Definite Integrals

Common Question Types and Typical Example Questions

Frequently Tested Contents

• (1) Improper integrals over an infinite interval
• (2) Improper integrals of unbounded functions

Common Question Types and Typical Examples

• Problem Type 1: Convergence and Divergence of Improper Integrals
• Problem Type 2: Calculation of Improper Integrals

1.1 Improper integrals of infinity

1.1.1 Basic Concepts

Introduction

• Basic conditions for definite integrals: $[a,b]$ is finite, and $f(x)$ is bounded;
• When an integral from $a$ to infinity is needed: $\int_{a}^{+\infty}f(x)dx$
• At this point, you can first set a $b$, then let b approach its limit;

#####Definition: Anomalousintegral

description: integral with infinite intervals or unbounded integrands is called improper integral; also known as generalized integral;

Explanation

• When it approaches infinity, if the limit has value, then it is called integral convergence;

Both the upper and lower realms are infinite

• Find: All values from negative infinity to positive infinity:
• $\int_{-\infty}^{+\infty}f(x)dx=\int_{-\infty}^{0}f(x)dx+\int_{0}^{+\infty}f(x)dx$
• Both limits must exist and converge; only then does the improper integral of the current upper and lower bounds converge;

Corollary: Generalized Newton-Leibniz Formula

• $\int_{-\infty}^{+\infty}f(x)dx=F(x)|_{-\infty}^{+\infty}$

#####Definition: Improperintegralsoveraninfiniteinterval

description:

1. Definition 1: $$\int_{a}^{+\infty}f(x)dx=\lim_{t\to+\infty}\int_{a}^{t}f(x)dx$$ If this limit exists, then the improper integral is called convergent; Conversely, it is divergent;

2. Definition 2: $$\int_{-\infty}^{b}f(x)dx=\lim_{t\to-\infty}\int_{t}^{b}f(x)dx$$ Same logic as definition one;

3. Definition 3: $$\int_{-\infty}^{+\infty}f(x)dx=\int_{-\infty}^{0}f(x)dx+\int_{0}^{+\infty}f(x)dx$$ If both converge, then converge; Otherwise, divergence;

Explanation

• Integral on an infinite interval, defined using the limit of the integral over a finite interval;
• Convergence can be evaluated;

1.1.2 Calculation Methods

#####Theorem: ComparativeDiscrimination

description: used to determine the divergence of improper integrals;

Suppose $f (x), g (x)$ continuity on $[a, +\infty)$, and $0\leq f (x)\leq g (x)$, then

$$1)\int_{a}^{+\infty}g(x)dx\text{ Convergence }\Rightarrow\int_{a}^{+\infty}f(x)dx\text{ Convergence }$$

$$2) \int_{a}^{+\infty}f(x)dx\text{ Diverges }\Rightarrow\int_{a}^{+\infty}g(x)dx\text{ Diverges }$$

Explanation

• Large convergence, small convergence; Large divergence, small divergence not necessarily divergence;
• Small things diverge, big ones definitely diverge; Small things converge, big ones don't necessarily converge;

Supplement: Preliminary Judgment on Zoom in and Out

• The number of times to observe the variable x in the function, for example, in $\frac{\sqrt{x}}{1+x^2}$ where the denominator x is $3/2$, and convergence;

Usage Examples

• Example question: $\int_1^{+\infty}\frac{\sqrt{x}}{1+x^2}dx.$
• Since $1+x^2>x^2$, $\frac{\sqrt{x}}{1+x^{2}}<\frac{\sqrt{x}}{x^{2}}=\frac{1}{x^{3/2}}$;
• Because the comparison integral has exponent $P=\frac32>1$, $\int_1^{+\infty}\frac{\sqrt{x}}{1+x^2}\,dx$ converges;

Theorem: LimitComparisonTest

Description:

Let $f,g$ be nonnegative and continuous on $[a,+\infty)$, and suppose $\lim_{x\to+\infty}\frac{f(x)}{g(x)}=\lambda$. Then:

1. If $\lambda>0$, then $\int_a^\infty f(x)\,dx$ and $\int_a^\infty g(x)\,dx$ have the same convergence behavior.

2. If $\lambda=0$ and $\int_a^\infty g(x)\,dx$ converges, then $\int_a^\infty f(x)\,dx$ converges.

3. If $\lambda=\infty$ and $\int_a^\infty g(x)\,dx$ diverges, then $\int_a^\infty f(x)\,dx$ diverges.

Common result: $$\int_{a}^{+\infty}\frac{1}{x^{P}}\,dx\begin{cases}\text{converges},&P>1,\\\text{diverges},&P\leq1,\end{cases}\qquad(a>0).$$

1.1.3 Example Problems

Example Question: Find $\int_{-\infty}^{+\infty}\frac{dx}{1+x^{2}}$

• Solution:
• $\arctan x\big|_{-\infty}^{+\infty}=\frac{\pi}{2}-(-\frac{\pi}{2})=\pi$;
• Question Type: ImproperIntegral

1.2 Improper integrals of unbounded functions

1.2.1 Basic Concepts

Definition: ImproperIntegralsOfUnboundedFunctions

Description:

1. If $f(x)$ is unbounded in every right-hand neighborhood of $a$, then $a$ is a singular point and $$\int_a^bf(x)\,dx=\lim_{t\to a^+}\int_t^bf(x)\,dx;$$

2. If $b$ is a singular point, then $$\int_{a}^{b}f(x)\,dx=\lim_{t\to b^{-}}\int_{a}^{t}f(x)\,dx;$$

3. If $c$ is a singular point with $a<c<b$, then both sides must converge and $$\int_{a}^{b}f(x)\,dx=\int_{a}^{c}f(x)\,dx+\int_{c}^{b}f(x)\,dx.$$

Theorem: ComparisonTest

Description:

Let $f,g$ be continuous on $(a,b]$ and satisfy $0\leq f(x)\leq g(x)$. Then:

$$\int_{a}^{b}g(x)\,dx\text{ converges }\Rightarrow\int_{a}^{b}f(x)\,dx\text{ converges};$$

$$\int_{a}^{b}f(x)\,dx\text{ diverges }\Rightarrow\int_{a}^{b}g(x)\,dx\text{ diverges}.$$

Theorem: LimitComparisonTest

Description:

Let $f,g$ be nonnegative and continuous on $(a,b]$, and suppose $\lim_{x\to a^+}\frac{f(x)}{g(x)}=\lambda$. Then

$$\begin{aligned}&1)\ \lambda>0:\ \int_a^b f(x)\,dx\text{ and }\int_a^b g(x)\,dx\text{ have the same convergence behavior};\\&2)\ \lambda=0:\ \int_a^b g(x)\,dx\text{ convergent}\Rightarrow\int_a^b f(x)\,dx\text{ convergent};\\&3)\ \lambda=+\infty:\ \int_a^b g(x)\,dx\text{ divergent}\Rightarrow\int_a^b f(x)\,dx\text{ divergent}.\end{aligned}$$

Common result: $$\int_a^b\frac{dx}{(x-a)^P},\quad\int_a^b\frac{dx}{(b-x)^P}\quad\begin{cases}\text{converge},&P<1,\\\text{diverge},&P\geq1.\end{cases}$$

Concept Introduction

• Assume the current function is $y=1/x$, then the current requirement is to integrate the function from 0 to 1;
• At point 0, since the limit is undefined, this point is also called a defect point;
• The imperfect integral of this unbounded function is also called the improper integral;

The Deceptive Nature of Flawed Points

• Only looking at the upper and lower limits of the definite integral cannot determine whether the current integral is a flawed integral;
• You need to check the current conditions for the function to see if they are met;

1.3 Frequently Tested Question Types

Question Type: Convergenceofimproperintegrals

PART 1: Problem-solving methods

Methods for Determining Convergence and Divergence

• 1. Definition method;
• Based on the current integrand, find its antiderivative, find its limit on the integrand interval, and see if the limit value converges;
• 2. Comparative discriminant;
• Comparative law;
• The limit form of the comparative method;
• 3. P Integral;
• Note:
• The infinite impermanent integral converges at -> $P>1$ and diverges at $P<1$
• The improper integral of an unbounded function converges at -> $P<1$ and diverges at $P>1$
• Use whichever is most convenient;

Note: When it comes to $\mathrm{e}^{\infty}$, pay attention to dividing it into left and right sections;

PART 2: Typical Example Problems

Example Question: $\int_{2}^{+\infty}\frac{1}{\sqrt{x}}dx$

• Analysis
• Method 1: Use definitions
• Method 2: Use a P-series
• Analysis
• Method One:
• $\int_{2}^{+\infty}\frac{1}{\sqrt{x}}dx$ Find the antiderivative -> which is 2√x at 2 times. At this point, directly find its limit;
• Method Two:
• Since the radical x is half of the P series, it diverges;
• Question Type: Convergenceanddivergenceofimproperintegrals

Example Question: $\text{ Anomalous integral }\int_{0}^{+\infty}\frac1{x^a(1+x)^b}dx{\text{ To converge means to contract }}$

• Analysis
• $\int_{0}^{+\infty}\frac{1}{x^{a}(i+x)^{b}}dx=\int_{0}^{1}\frac{1}{x^{a}(i+x)^{b}}dx+\int_{1}^{ts}\frac{1}{x^{a}(i+x)^{b}}dx$
• Both need to converge;
• Analysis of the left side:
• Use the comparative method: $\lim_{x\to0^{+}}\frac{\frac{1}{x^{a}(x+x)^{b}}}{\frac{1}{x^{a}}}=1$
• Therefore, the convergence and divergence with the P integral -> $a<1$
• Analysis of the right side:
• $\frac1{x^{a+b}(1+\frac1x)^b}$
• Because it approaches infinity, 1+1/x equals 1, so only $a+b$ is needed
• Limit form using the comparative method: $\lim_{x\to\infty}\frac{\frac1{x^{a}(x)^{b}}}{\frac1{x^{a+b}}}$ -> $a+b>1$
• Analysis
• Question Type: #

PART 3: Key Points Review

Question Type: Calculationofimproperintegrals

PART 1: Problem-solving methods

What is the Calculation of Improper Integrals

• Definition:
• Calculate a definite integral + calculate a limit;
• Method:
• Method 1: Substitution method
• Method 2: Partial Method

PART 2: Typical Example Problems

Example Question: $\int_{2}^{+\infty}\frac{\mathrm{d}x}{(x+7)\sqrt{x-2}}=$

• Analysis
• Solution 1: Substitution method
• Ling $\sqrt{k-2}=t,\quad x-2=t^{2}$
• So: $\int_{0}^{+\infty}\frac{2t}{t(t^{2}+9)}dt=2\int_{0}^{+\infty}\frac{dt}{9+t^{2}}$
• $=\frac{2}{3}\arctan\frac{\pi}{3}|_{0}^{+\infty}=\frac{2}{3}\left[\frac{\pi}{2}-0\right]$
• Solution Two: Completing the differential
• Original expression = $\int_{2}^{+\infty}\frac{2d\sqrt{x-2}}{9+(\sqrt{x-2})^{2}}$

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• Analysis
• Question Type: #

PART 3: Key Points Review