Problem. Let R be the region bounded above by z = 8 - x2 – y? and below by z = x² + y?. Then the volume of R is:
Problem. Let R be the region bounded above by z = 8 - x2 – y? and below by z = x² + y?. Then the volume of R is:
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 1:**
Let \( R \) be the region bounded above by \( z = 8 - x^2 - y^2 \) and below by \( z = x^2 + y^2 \). Then the volume of \( R \) is: \(\boxed{?} \)
**Problem 2:**
Let \( R \) be the portion of \( x^2 + y^2 + z^2 = 4 \) above the plane \( z=1 \). Consider the integral \(\iiint_R z \, dV \).
- **In cylindrical coordinates, the integral would be:**
\[
\int_{\boxed{?}}^{\boxed{?}} \int_{\boxed{?}}^{\boxed{?}} \int_{\boxed{?}}^{\boxed{?}} z \, dz \, dr \, d\theta.
\]
- **In spherical coordinates, the integral would be (input "p" for phi):**
\[
\int_{\boxed{?}}^{\boxed{?}} \int_{\boxed{?}}^{\boxed{?}} \int_{\boxed{?}}^{\boxed{?}} \rho^3 \cos(p) \sin(p) \, d\rho \, dp \, d\theta.
\]
- **Evaluating either way gives an answer of:** \(\boxed{?} \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25fb0993-5d26-4149-a829-39bf7f0bd1d0%2Fdee9af19-f6a0-4aa7-8251-0bc88661ed5c%2Flvi6tc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 1:**
Let \( R \) be the region bounded above by \( z = 8 - x^2 - y^2 \) and below by \( z = x^2 + y^2 \). Then the volume of \( R \) is: \(\boxed{?} \)
**Problem 2:**
Let \( R \) be the portion of \( x^2 + y^2 + z^2 = 4 \) above the plane \( z=1 \). Consider the integral \(\iiint_R z \, dV \).
- **In cylindrical coordinates, the integral would be:**
\[
\int_{\boxed{?}}^{\boxed{?}} \int_{\boxed{?}}^{\boxed{?}} \int_{\boxed{?}}^{\boxed{?}} z \, dz \, dr \, d\theta.
\]
- **In spherical coordinates, the integral would be (input "p" for phi):**
\[
\int_{\boxed{?}}^{\boxed{?}} \int_{\boxed{?}}^{\boxed{?}} \int_{\boxed{?}}^{\boxed{?}} \rho^3 \cos(p) \sin(p) \, d\rho \, dp \, d\theta.
\]
- **Evaluating either way gives an answer of:** \(\boxed{?} \)
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