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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![On this page, we focus on a double integral notation typically covered in multivariable calculus:
\[
\int_{-1}^{1} \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} (x^2 + y^2)^{3/2} \, dy \, dx
\]
### Explanation:
- The outer integral (with respect to \(x\)) ranges from \(-1\) to \(1\).
- The inner integral (with respect to \(y\)) is bounded by \(-\sqrt{1-x^2}\) to \(\sqrt{1-x^2}\), indicating the region under which the function is integrated.
- The integrand, \((x^2 + y^2)^{3/2}\), represents the function to be integrated over the described region.
This setup is typical for evaluating integrals over circular or elliptical regions since the inner bounds suggest a circle due to the \(\sqrt{1-x^2}\) term, corresponding to a semicircular cross-section in the xy-plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9f391b6-5f04-4d98-9bc5-d57c88046352%2Fe5efe8d4-7286-47dd-a765-bfee33df977a%2F7fsg074_processed.jpeg&w=3840&q=75)
Transcribed Image Text:On this page, we focus on a double integral notation typically covered in multivariable calculus:
\[
\int_{-1}^{1} \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} (x^2 + y^2)^{3/2} \, dy \, dx
\]
### Explanation:
- The outer integral (with respect to \(x\)) ranges from \(-1\) to \(1\).
- The inner integral (with respect to \(y\)) is bounded by \(-\sqrt{1-x^2}\) to \(\sqrt{1-x^2}\), indicating the region under which the function is integrated.
- The integrand, \((x^2 + y^2)^{3/2}\), represents the function to be integrated over the described region.
This setup is typical for evaluating integrals over circular or elliptical regions since the inner bounds suggest a circle due to the \(\sqrt{1-x^2}\) term, corresponding to a semicircular cross-section in the xy-plane.
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