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(c) Consider the double integral || xy dA where D is bounded by the x-axis and the upper half semicircles of two circles x? + y? = 4 and x? +y? = 1; see the figure below.

(c) Consider the double integral || xy dA where D is bounded by the x-axis and the upper half semicircles of two circles x? + y? = 4 and x? +y? = 1; see the figure below.

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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(c) Consider the double integral || xy dA where D is bounded by the x-axis and the upper half semicircles of two circles x? + y? = 4 and x? +y? = 1; see the figure below. x² + y? = 4 12²+ y² = 1 -2 -1 1 2 We may describe D in polar coordinates D = { (r, 0) | 0 <o < T,1 < r < 2 } and obtain that 2 xy dA = I/ (r cos 0) (r sin 0)rdrdð. Alternatively, we may also describe D by dividing it to three separate regions D1, D2, and D3, where D1, D2, and D3 can be described by rectangular coordinates D; = { (r, 3) | –2 <= <-1,0 < y< V4 – a² }, D2 ={ (x, y) –1 < x < 1, v1 – x² < y< V4- x² } , and D3 = { (x, y) | 1 < x < 2,0 < y < v4 – x² .
Transcribed Image Text:(c) Consider the double integral || xy dA where D is bounded by the x-axis and the upper half semicircles of two circles x? + y? = 4 and x? +y? = 1; see the figure below. x² + y? = 4 12²+ y² = 1 -2 -1 1 2 We may describe D in polar coordinates D = { (r, 0) | 0 <o < T,1 < r < 2 } and obtain that 2 xy dA = I/ (r cos 0) (r sin 0)rdrdð. Alternatively, we may also describe D by dividing it to three separate regions D1, D2, and D3, where D1, D2, and D3 can be described by rectangular coordinates D; = { (r, 3) | –2 <= <-1,0 < y< V4 – a² }, D2 ={ (x, y) –1 < x < 1, v1 – x² < y< V4- x² } , and D3 = { (x, y) | 1 < x < 2,0 < y < v4 – x² .
Consequently, we will have -1 V4-x² 4–x² xy dA = xy dydx + xy dydx + xy dydx V1-x²
Transcribed Image Text:Consequently, we will have -1 V4-x² 4–x² xy dA = xy dydx + xy dydx + xy dydx V1-x²
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