Use the given transformation to evaluate the given integral, where R is the triangular region with vertices (0, 0), (6, 1), and (1, 6). (x - 3y) dA, x = 6u + v, y = u + 6v
Use the given transformation to evaluate the given integral, where R is the triangular region with vertices (0, 0), (6, 1), and (1, 6). (x - 3y) dA, x = 6u + v, y = u + 6v
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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![Use the given transformation to evaluate the given integral, where R is the triangular region with vertices (0,
0), (6, 1), and (1, 6).
(x - 3y) dA, x = 6u + v, y = u + 6v
Step 1
For the transformation x = 6u + v, y =u + 6v, the Jacobian is
1
a(x, y).
a(u, v)
ди
dv
= 35
ду ду
1
6.
du
dv
Also,
X - 3y = (6u + v) - 3(u + 6v) = 3u -
17
v.
Step 2
To find the region S in the uv-plane which corresponds to R, we find the corresponding boundaries. The line
though (0, 0) and (6, 1) is y = 1/6
x, and this is the image of v = 0
Step 3
The line through (6, 1) and (1, 6) is y =
and this is the image of v =
The line through (0, 0) and (1, 6) is y =
and this is the image of u = 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9f8f1aa0-ff42-403f-89a3-b6ebbbc6b6ad%2F7b0769e9-2c27-477a-b91a-32dc6dd073f2%2Fzjd4blu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Use the given transformation to evaluate the given integral, where R is the triangular region with vertices (0,
0), (6, 1), and (1, 6).
(x - 3y) dA, x = 6u + v, y = u + 6v
Step 1
For the transformation x = 6u + v, y =u + 6v, the Jacobian is
1
a(x, y).
a(u, v)
ди
dv
= 35
ду ду
1
6.
du
dv
Also,
X - 3y = (6u + v) - 3(u + 6v) = 3u -
17
v.
Step 2
To find the region S in the uv-plane which corresponds to R, we find the corresponding boundaries. The line
though (0, 0) and (6, 1) is y = 1/6
x, and this is the image of v = 0
Step 3
The line through (6, 1) and (1, 6) is y =
and this is the image of v =
The line through (0, 0) and (1, 6) is y =
and this is the image of u = 0.
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