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posted by  abcd12345 on 11/4/2009 11:44:19 PM  |  status: Closed  |  Earned Karma: 48

Double integrals - Volume

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Calculus N/A N/A N/A 11/4/2009 at 11:00:00 PM
Question Details:
Find the volume of the region lying under the graph of the function f(x,y)=cos(x^2+y^2)+1, which lies over the circle of radius 3 in the x-y plane centered at the origin.
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posted by anonymousxyz on 11/5/2009 12:56:47 AM  |  status: Live
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Response Details:


\text{Use polar coordinates to evaluate the integral:}


f( x, y ) = \cos( x^2 + y^2 ) + 1 = \cos( r^2 ) + 1


\begin{align*} V &= \int \int_R f(x, y) \;\text{d}A \\\\ &= \int_0^{2 \pi} \int_0^3 \big[ \cos (r^2) + 1 \big] \; r\, \text{d}r \,\text{d}\theta \\\\ &= \int_0^{2 \pi} \int_0^3 \big[ r \cos (r^2) + r \big] \, \text{d}r \,\text{d}\theta \\\\ &= \int_0^{2 \pi} \frac{\sin (r^2) + r^2}{2} \;\bigg|_0^3 \;\text{d}\theta \\\\ &= \int_0^{2 \pi} \frac{\sin 9 + 9}{2}\;\text{d}\theta \\\\ &= \bigg[\frac{\sin 9 + 9}{2}\bigg]\theta\;\bigg|_0^{2 \pi} \end{align*}


      \\\\ &= \big[ \sin 9 + 9 \big]\pi
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