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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

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
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.
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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