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
![**Apply the substitution using steps 1 and 2 and rewrite the integral in terms of θ.**
\[
\int \sqrt{16 - 6x^2} \, dx = \int (4 \cos \theta) \left(\frac{4}{\sqrt{6}} \cos \theta \, d\theta \right)
\]
\[
= \frac{16}{\sqrt{6}} \int \, \boxed{\phantom{xxxx}} \, d\theta
\]
### Explanation:
The expression begins with an integral \(\int \sqrt{16 - 6x^2} \, dx\), which is transformed using substitution involving trigonometric identities.
- **Step 1** involves expressing the integral in terms of \( \theta \) using the substitution \( x = \frac{4}{\sqrt{6}} \sin \theta \). Substituting in, you find that \( \cos \theta \) expressions appear in the integrand.
- **Step 2** involves simplifying any constants and terms under the integral. The integral \(\int (4 \cos \theta) \left(\frac{4}{\sqrt{6}} \cos \theta \, d\theta \right)\) shows those expressions, leading to factors outside the integral as \(\frac{16}{\sqrt{6}} \int \, \boxed{\phantom{xxxx}} \, d\theta\).
The blank box represents the expression that would result from completing this conversion process, focusing on \( \theta \) variables.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcdc0c33c-1cee-4754-904a-1b31674d8af7%2F371d76ef-a298-4038-a4d6-89bf91820220%2Fxfaa43n_processed.png&w=3840&q=75)
Transcribed Image Text:**Apply the substitution using steps 1 and 2 and rewrite the integral in terms of θ.**
\[
\int \sqrt{16 - 6x^2} \, dx = \int (4 \cos \theta) \left(\frac{4}{\sqrt{6}} \cos \theta \, d\theta \right)
\]
\[
= \frac{16}{\sqrt{6}} \int \, \boxed{\phantom{xxxx}} \, d\theta
\]
### Explanation:
The expression begins with an integral \(\int \sqrt{16 - 6x^2} \, dx\), which is transformed using substitution involving trigonometric identities.
- **Step 1** involves expressing the integral in terms of \( \theta \) using the substitution \( x = \frac{4}{\sqrt{6}} \sin \theta \). Substituting in, you find that \( \cos \theta \) expressions appear in the integrand.
- **Step 2** involves simplifying any constants and terms under the integral. The integral \(\int (4 \cos \theta) \left(\frac{4}{\sqrt{6}} \cos \theta \, d\theta \right)\) shows those expressions, leading to factors outside the integral as \(\frac{16}{\sqrt{6}} \int \, \boxed{\phantom{xxxx}} \, d\theta\).
The blank box represents the expression that would result from completing this conversion process, focusing on \( \theta \) variables.
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