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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Question
![**Problem Statement:**
Use the substitution \( x = 7 \sin t \) to evaluate the integral \( \int \sqrt{49 - x^2} \, dx \).
**Approach:**
To solve this integral, we perform the substitution \( x = 7 \sin t \). This substitution is useful for integrals involving the square root of the difference of squares, which often simplifies the integrand into a trigonometric identity.
- When \( x = 7 \sin t \), then \( dx = 7 \cos t \, dt \).
- The expression under the square root becomes \( 49 - x^2 = 49 - 49 \sin^2 t = 49(1 - \sin^2 t) = 49 \cos^2 t \).
**Integration Steps:**
1. Substitute \( x = 7 \sin t \) and \( dx = 7 \cos t \, dt \) into the integral:
\[
\int \sqrt{49 - x^2} \, dx = \int \sqrt{49 \cos^2 t} \cdot 7 \cos t \, dt
\]
2. Simplify the integrand:
\[
\int 7 \cos t \cdot 7 \cos t \, dt = \int 49 \cos^2 t \, dt
\]
3. Perform the integration using trigonometric identities or standard integral results for \( \cos^2 t \).
**Final Result:**
By evaluating the integral and substituting back in terms of \( x \), you will find the solution to the original integral problem.
This type of substitution helps simplify complex integrals using trigonometric identities and transformations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd4f4b624-9180-41b5-94fa-c093ec7455d7%2F7565ea01-a568-4aa2-9184-89e21b9ae911%2Fsyd63hel_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Use the substitution \( x = 7 \sin t \) to evaluate the integral \( \int \sqrt{49 - x^2} \, dx \).
**Approach:**
To solve this integral, we perform the substitution \( x = 7 \sin t \). This substitution is useful for integrals involving the square root of the difference of squares, which often simplifies the integrand into a trigonometric identity.
- When \( x = 7 \sin t \), then \( dx = 7 \cos t \, dt \).
- The expression under the square root becomes \( 49 - x^2 = 49 - 49 \sin^2 t = 49(1 - \sin^2 t) = 49 \cos^2 t \).
**Integration Steps:**
1. Substitute \( x = 7 \sin t \) and \( dx = 7 \cos t \, dt \) into the integral:
\[
\int \sqrt{49 - x^2} \, dx = \int \sqrt{49 \cos^2 t} \cdot 7 \cos t \, dt
\]
2. Simplify the integrand:
\[
\int 7 \cos t \cdot 7 \cos t \, dt = \int 49 \cos^2 t \, dt
\]
3. Perform the integration using trigonometric identities or standard integral results for \( \cos^2 t \).
**Final Result:**
By evaluating the integral and substituting back in terms of \( x \), you will find the solution to the original integral problem.
This type of substitution helps simplify complex integrals using trigonometric identities and transformations.
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