Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Finding Definite and Improper Integrals**
**Problem:**
Evaluate the following definite, improper integral:
\[
\int_{x=1}^{+\infty} \frac{1}{x^4} \, dx
\]
**Explanation:**
To solve this integral, follow these steps:
1. Recognize that this is an improper integral because it involves an infinite limit of integration.
2. Rewrite the integral with a limit to handle the infinity.
\[
\int_{x=1}^{+\infty} \frac{1}{x^4} \, dx = \lim_{b \to \infty} \int_{x=1}^{b} \frac{1}{x^4} \, dx
\]
3. Evaluate the integral on the interval \([1, b]\):
\[
\int_{x=1}^{b} \frac{1}{x^4} \, dx
\]
4. Perform the integration:
\[
= \left[ -\frac{1}{3x^3} \right]_{1}^{b}
\]
5. Substitute the bounds into the evaluated integral:
\[
= \lim_{b \to \infty} \left( -\frac{1}{3b^3} + \frac{1}{3 \cdot 1^3} \right)
\]
6. Simplify the expression:
\[
= \lim_{b \to \infty} \left( -\frac{1}{3b^3} + \frac{1}{3} \right)
\]
7. As \( b \to \infty \), \(\frac{1}{3b^3} \to 0\), so the expression converges to:
\[
= 0 + \frac{1}{3} = \frac{1}{3}
\]
Therefore,
\[
\int_{x=1}^{+\infty} \frac{1}{x^4} \, dx = \frac{1}{3}
\]
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F239df334-5b03-4bc0-9001-d398335a6cd3%2F60646cc6-0b70-4557-9925-4a59ce53c1ab%2F6ziqrck_reoriented.jpeg&w=3840&q=75)
Transcribed Image Text:---
**Finding Definite and Improper Integrals**
**Problem:**
Evaluate the following definite, improper integral:
\[
\int_{x=1}^{+\infty} \frac{1}{x^4} \, dx
\]
**Explanation:**
To solve this integral, follow these steps:
1. Recognize that this is an improper integral because it involves an infinite limit of integration.
2. Rewrite the integral with a limit to handle the infinity.
\[
\int_{x=1}^{+\infty} \frac{1}{x^4} \, dx = \lim_{b \to \infty} \int_{x=1}^{b} \frac{1}{x^4} \, dx
\]
3. Evaluate the integral on the interval \([1, b]\):
\[
\int_{x=1}^{b} \frac{1}{x^4} \, dx
\]
4. Perform the integration:
\[
= \left[ -\frac{1}{3x^3} \right]_{1}^{b}
\]
5. Substitute the bounds into the evaluated integral:
\[
= \lim_{b \to \infty} \left( -\frac{1}{3b^3} + \frac{1}{3 \cdot 1^3} \right)
\]
6. Simplify the expression:
\[
= \lim_{b \to \infty} \left( -\frac{1}{3b^3} + \frac{1}{3} \right)
\]
7. As \( b \to \infty \), \(\frac{1}{3b^3} \to 0\), so the expression converges to:
\[
= 0 + \frac{1}{3} = \frac{1}{3}
\]
Therefore,
\[
\int_{x=1}^{+\infty} \frac{1}{x^4} \, dx = \frac{1}{3}
\]
---
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