Consider the double integal where Dis the semiciroular region given by (x. y) |x + s4 and y 2 0). (a) Aher converting the doule integral above to an iterated integral, one obtaine of ydy dx Etydy da. L tydyds. LEydy ds. of LEydy dx. Aher taking the terated integral from part (a) and changing the order of integration, one obtains x'yds dy. x*ydx dy. f ydr dy. f LEyds dy iyds dy. f ydr dy () Aher comverting the double integral above to an terated integral in polar coordinates, one obtains of" con' e sin de de. of" sin' cos dr de of con sin dr de. f r con e sin edr de. Of sinecosdr de. of ir sin' o cos e dr de. of" sin' cos dr de of" con' e sin dr de ( The value of the double integral above can be expressed as the fraction in lowest terms, where a
Consider the double integal where Dis the semiciroular region given by (x. y) |x + s4 and y 2 0). (a) Aher converting the doule integral above to an iterated integral, one obtaine of ydy dx Etydy da. L tydyds. LEydy ds. of LEydy dx. Aher taking the terated integral from part (a) and changing the order of integration, one obtains x'yds dy. x*ydx dy. f ydr dy. f LEyds dy iyds dy. f ydr dy () Aher comverting the double integral above to an terated integral in polar coordinates, one obtains of" con' e sin de de. of" sin' cos dr de of con sin dr de. f r con e sin edr de. Of sinecosdr de. of ir sin' o cos e dr de. of" sin' cos dr de of" con' e sin dr de ( The value of the double integral above can be expressed as the fraction in lowest terms, where a
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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Transcribed Image Text:Consider the double integral
x²y dA,
where D is the semicircular region given by {(x, y) | x + y < 4 and y 2 0}.
(a) After converting the double integral above to an iterated integral, one obtains
x²y dy dx
of VE x²y dy dx.
x²y dy dx.
of Vxy dy dx.
x²y dy dx.
x²y dy dx.
x²y dy dx.
of LEx'y dy dx.
(b) After taking the iterated integral from part (a) and changing the order of integration, one obtains
x²y dx dy.
f LE x*y dx dy.
o VA- x*y dx dy.
x*y dx dy.
x²y dx dy.
V4-y
x²ydx dy-
ofLEry dx dy.
o VE x?y dx dy.
(c) After converting the double integral above to an iterated integral in polar coordinates, one obtains
o* G cos e sin ® dr de.
o* G sin? 0 cos 0 dr dº.
of L cos? 0 sin 0 dr d0.
o* cos? 0 sin e dr de.
of" L rA sin? 0 cos 0 dr d0.
of L r sin? 0 cos 0 dr de.
of* r sin? 0 cos e dr de.
o* G" cos 0 sin 0 dr d0.
(d) The value of the double integral above can be expressed as the fraction - in lowest terms, where a =
and b
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