To evaluate the following integral, carry out these steps. a. Sketch the original region of integration R in the xy-plane and the new region S in the uv-plane using the given change of variables. b. Find the limits of integration for the new integral with respect to u and v. c. Compute the Jacobian. d. Change variables and evaluate the new integral. X SSx²√x + 7y dA, where R = {(x,y): 0≤x≤7, - ≤ y ≤6-x}; use x = 7u, R y = 6v-u. b. Find the limits of integration. 0 sus 1 0 ≤v≤ 1-u c. J(u, v) 42 (Simplify your answer.) d. [√x² √x + 7ydA= SSx². [ R
To evaluate the following integral, carry out these steps. a. Sketch the original region of integration R in the xy-plane and the new region S in the uv-plane using the given change of variables. b. Find the limits of integration for the new integral with respect to u and v. c. Compute the Jacobian. d. Change variables and evaluate the new integral. X SSx²√x + 7y dA, where R = {(x,y): 0≤x≤7, - ≤ y ≤6-x}; use x = 7u, R y = 6v-u. b. Find the limits of integration. 0 sus 1 0 ≤v≤ 1-u c. J(u, v) 42 (Simplify your answer.) d. [√x² √x + 7ydA= SSx². [ R
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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![To evaluate the following integral, carry out these steps:
**a.** Sketch the original region of integration \( R \) in the xy-plane and the new region \( S \) in the uv-plane using the given change of variables.
**b.** Find the limits of integration for the new integral with respect to \( u \) and \( v \).
**c.** Compute the Jacobian.
**d.** Change variables and evaluate the new integral.
\[
\iint_R x^2 \sqrt{x + 7y} \, dA, \quad \text{where } R = \left\{ (x,y): 0 \leq x \leq 7, \, -\frac{x}{7} \leq y \leq 6 - x \right\}; \, \text{use } x = 7u, \, y = 6v - u.
\]
---
**b.** Find the limits of integration.
\[
0 \leq u \leq 1
\]
\[
0 \leq v \leq 1 - u
\]
**c.** \( J(u,v) = 42 \) (Simplify your answer.)
**d.** \[
\iint_R x^2 \sqrt{x + 7y} \, dA = \boxed{}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff237bbf8-6485-48b1-b1a9-4cbf0285c93d%2F6cf78aa4-2868-42f1-9a21-4667dbcd5607%2F6p3zdo_processed.png&w=3840&q=75)
Transcribed Image Text:To evaluate the following integral, carry out these steps:
**a.** Sketch the original region of integration \( R \) in the xy-plane and the new region \( S \) in the uv-plane using the given change of variables.
**b.** Find the limits of integration for the new integral with respect to \( u \) and \( v \).
**c.** Compute the Jacobian.
**d.** Change variables and evaluate the new integral.
\[
\iint_R x^2 \sqrt{x + 7y} \, dA, \quad \text{where } R = \left\{ (x,y): 0 \leq x \leq 7, \, -\frac{x}{7} \leq y \leq 6 - x \right\}; \, \text{use } x = 7u, \, y = 6v - u.
\]
---
**b.** Find the limits of integration.
\[
0 \leq u \leq 1
\]
\[
0 \leq v \leq 1 - u
\]
**c.** \( J(u,v) = 42 \) (Simplify your answer.)
**d.** \[
\iint_R x^2 \sqrt{x + 7y} \, dA = \boxed{}
\]
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