d. Change variables and evaluate the new integral. SSx²√x + 7y dA, where R= R y = 6v-u. X 0≤x≤7,-≤y≤6- where R=(x,y): 0≤x≤7, {(x,y): >; use x = 7u, u
d. Change variables and evaluate the new integral. SSx²√x + 7y dA, where R= R y = 6v-u. X 0≤x≤7,-≤y≤6- where R=(x,y): 0≤x≤7, {(x,y): >; use x = 7u, u
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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Need help with d.
![# Evaluating a Double Integral with Change of Variables
To evaluate the following integral, carry out these steps:
### a. Sketch the Original Region and the New Region
- **Original Region in the xy-plane**: \( R = \{ (x,y): 0 \leq x \leq 7, -\frac{x}{7} \leq y \leq 6-x \} \)
- **New Region in the uv-plane**: Use the change of variables \( x = 7u \) and \( y = 6v - u \).
### b. Find the Limits of Integration
- For \( u \): \( 0 \leq u \leq 1 \)
- For \( v \): \( 0 \leq v \leq 1-u \)
### c. Compute the Jacobian
- **Jacobian**: \( J(u,v) = 42 \)
### d. Change Variables and Evaluate the New Integral
- Transform the integral with the change of variables and compute:
\[
\int \int_R x^2 \sqrt{x + 7y} \, dA = \text{Evaluate with new limits}
\]
- Rectangle or blanks indicate areas to insert specific numeric values or further computations.
This task involves understanding region transformations and computing integrals over transformed boundaries using the Jacobian determinant for variable changes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff237bbf8-6485-48b1-b1a9-4cbf0285c93d%2F4c2a092d-5ca9-4691-ad4b-c05f57620c9c%2F4ri0e_processed.png&w=3840&q=75)
Transcribed Image Text:# Evaluating a Double Integral with Change of Variables
To evaluate the following integral, carry out these steps:
### a. Sketch the Original Region and the New Region
- **Original Region in the xy-plane**: \( R = \{ (x,y): 0 \leq x \leq 7, -\frac{x}{7} \leq y \leq 6-x \} \)
- **New Region in the uv-plane**: Use the change of variables \( x = 7u \) and \( y = 6v - u \).
### b. Find the Limits of Integration
- For \( u \): \( 0 \leq u \leq 1 \)
- For \( v \): \( 0 \leq v \leq 1-u \)
### c. Compute the Jacobian
- **Jacobian**: \( J(u,v) = 42 \)
### d. Change Variables and Evaluate the New Integral
- Transform the integral with the change of variables and compute:
\[
\int \int_R x^2 \sqrt{x + 7y} \, dA = \text{Evaluate with new limits}
\]
- Rectangle or blanks indicate areas to insert specific numeric values or further computations.
This task involves understanding region transformations and computing integrals over transformed boundaries using the Jacobian determinant for variable changes.
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