Evaluate the triple integral u= 2x-y 2 v= / 1 [ F(z, y, z)dV = [ ] [ H(u, v, w)| J(u, v, w)|dudvdu F Remember that: +5 CCT** f(x, y, z)dxdydz where f(x, y, z) = 1 + 0 70 and w= 흉 Triple Integral Region R 14 lower limit 16 upper limit= Ⓒlower limit= upper limit lower limit upper limit= # H(u, v, w) |J(u, v, w)| : [[[H(u, v, w).J(u, v, w)|dudvdu = 1102
Evaluate the triple integral u= 2x-y 2 v= / 1 [ F(z, y, z)dV = [ ] [ H(u, v, w)| J(u, v, w)|dudvdu F Remember that: +5 CCT** f(x, y, z)dxdydz where f(x, y, z) = 1 + 0 70 and w= 흉 Triple Integral Region R 14 lower limit 16 upper limit= Ⓒlower limit= upper limit lower limit upper limit= # H(u, v, w) |J(u, v, w)| : [[[H(u, v, w).J(u, v, w)|dudvdu = 1102
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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![### Evaluating Triple Integrals with Change of Variables
#### Problem Statement:
Evaluate the triple integral
\[
\iiint\limits_R \iiint\limits_0^5 \iiint\limits_y^4 \iiint\limits_{z/3}^{z/3 + 5} f(x, y, z) dz dy dz \quad \text{where} \; f(x, y, z) = x + \frac{z}{3}.
\]
Given transformations:
\[
u = \frac{2x - y}{2}, \quad v = \frac{y}{2}, \quad \text{and} \quad w = \frac{z}{3}.
\]
#### Triple Integral Region:
The region \(R\) is represented in a 3D plot with axes \(x\), \(y\), and \(z\). The bounds for \(x\), \(y\), and \(z\) are represented as different colored planes intersecting to form a prism-shaped volume.
#### Method:
Remember that:
\[
\iiint\limits_R f (x, y, z) dV = \iiint\limits_G H(u, v, w) |J(u, v, w)| du dv dw,
\]
where \(J(u, v, w)\) is the Jacobian determinant of the transformation.
#### Steps:
1. **Define the limits for the new variables \(u\), \(v\), and \(w\)**:
- \(u\) lower limit =
- \(u\) upper limit =
- \(v\) lower limit =
- \(v\) upper limit =
- \(w\) lower limit =
- \(w\) upper limit =
2. **Determine the transformed function \(H(u, v, w)\)**:
\[
H(u, v, w) =
\]
3. **Calculate the Jacobian determinant \( |J(u, v, w)| \)**:
\[
|J(u, v, w)| =
\]
4. **Set up the triple integral in terms of \(u\), \(v\), and \(w\)**:
\[
\iiint\limits_G H(u, v, w) |J(u, v, w)| du dv dw =
\]
If](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F74b6836e-7075-4604-8b7e-b1441272473c%2Fcbb51c57-39b9-42b0-a535-b19323c762fd%2Funhprn_processed.png&w=3840&q=75)
Transcribed Image Text:### Evaluating Triple Integrals with Change of Variables
#### Problem Statement:
Evaluate the triple integral
\[
\iiint\limits_R \iiint\limits_0^5 \iiint\limits_y^4 \iiint\limits_{z/3}^{z/3 + 5} f(x, y, z) dz dy dz \quad \text{where} \; f(x, y, z) = x + \frac{z}{3}.
\]
Given transformations:
\[
u = \frac{2x - y}{2}, \quad v = \frac{y}{2}, \quad \text{and} \quad w = \frac{z}{3}.
\]
#### Triple Integral Region:
The region \(R\) is represented in a 3D plot with axes \(x\), \(y\), and \(z\). The bounds for \(x\), \(y\), and \(z\) are represented as different colored planes intersecting to form a prism-shaped volume.
#### Method:
Remember that:
\[
\iiint\limits_R f (x, y, z) dV = \iiint\limits_G H(u, v, w) |J(u, v, w)| du dv dw,
\]
where \(J(u, v, w)\) is the Jacobian determinant of the transformation.
#### Steps:
1. **Define the limits for the new variables \(u\), \(v\), and \(w\)**:
- \(u\) lower limit =
- \(u\) upper limit =
- \(v\) lower limit =
- \(v\) upper limit =
- \(w\) lower limit =
- \(w\) upper limit =
2. **Determine the transformed function \(H(u, v, w)\)**:
\[
H(u, v, w) =
\]
3. **Calculate the Jacobian determinant \( |J(u, v, w)| \)**:
\[
|J(u, v, w)| =
\]
4. **Set up the triple integral in terms of \(u\), \(v\), and \(w\)**:
\[
\iiint\limits_G H(u, v, w) |J(u, v, w)| du dv dw =
\]
If
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