Evaluate ∫∫Rcos(x^2+y^2)dA, where R is the region above the x-axis within the circle x^2+y^2= 25.
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Expert's answer
2020-11-05T10:33:18-0500
We convert to polar co-ordinatesdA=rdrdθx=rcosθy=rsinθx2+y2=r2SinceRis the region above thex−axis,we only consider the first andsecond quadrants, and thus evaluatethe double integral.θis from0toπris from0to5∴∬Rcos(x2+y2)dA=∫05∫0πcos(x2+y2)rdθdr=π∫05cos(r2)rdr=2πsin(r2)∣∣05=2πsin(25)The area of the region is2πsin(25)
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