2 ² J z=0x= -√4-2² 4-x²-2 y=0 N dydxdz
Advanced Engineering Mathematics
10th Edition
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Convert the following triple integral to cylindrical coordinates or spherical coordinates (do NOT evaluate):
![This image shows a triple integral used in multivariable calculus, possibly to find the volume under a certain surface. The integral is presented as follows:
\[
\int_{z=0}^{2} \int_{x=-\sqrt{4-z^2}}^{\sqrt{4-z^2}} \int_{y=0}^{\sqrt{4-x^2-z^2}} z \, dy \, dx \, dz =
\]
Here’s a breakdown of the integral:
1. **The First Integral \( \int_{z=0}^{2} \)**:
- The outermost integral is with respect to \( z \) and it spans from 0 to 2.
2. **The Second Integral \( \int_{x=-\sqrt{4-z^2}}^{\sqrt{4-z^2}} \)**:
- The range of \( x \) depends on \( z \) and goes from \( -\sqrt{4-z^2} \) to \( \sqrt{4-z^2} \).
3. **The Third Integral \( \int_{y=0}^{\sqrt{4-x^2-z^2}} \)**:
- The range of \( y \) also depends on \( x \) and \( z \), going from 0 to \( \sqrt{4-x^2-z^2} \).
4. **The Integrand**:
- The integrand is \( z \), which is the function being integrated with respect to \( y \), \( x \), and \( z \).
This integral describes the volume enclosed by a certain surface or solid in a three-dimensional space, where the bounds for each variable depend on the others, suggesting a specific region of space being considered.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff930c139-ad70-48e2-a90a-c5db80131496%2F01dc430a-49ff-472c-809b-f15c9e1a0962%2F1o69g8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:This image shows a triple integral used in multivariable calculus, possibly to find the volume under a certain surface. The integral is presented as follows:
\[
\int_{z=0}^{2} \int_{x=-\sqrt{4-z^2}}^{\sqrt{4-z^2}} \int_{y=0}^{\sqrt{4-x^2-z^2}} z \, dy \, dx \, dz =
\]
Here’s a breakdown of the integral:
1. **The First Integral \( \int_{z=0}^{2} \)**:
- The outermost integral is with respect to \( z \) and it spans from 0 to 2.
2. **The Second Integral \( \int_{x=-\sqrt{4-z^2}}^{\sqrt{4-z^2}} \)**:
- The range of \( x \) depends on \( z \) and goes from \( -\sqrt{4-z^2} \) to \( \sqrt{4-z^2} \).
3. **The Third Integral \( \int_{y=0}^{\sqrt{4-x^2-z^2}} \)**:
- The range of \( y \) also depends on \( x \) and \( z \), going from 0 to \( \sqrt{4-x^2-z^2} \).
4. **The Integrand**:
- The integrand is \( z \), which is the function being integrated with respect to \( y \), \( x \), and \( z \).
This integral describes the volume enclosed by a certain surface or solid in a three-dimensional space, where the bounds for each variable depend on the others, suggesting a specific region of space being considered.
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