# Mass of Complicated 3D Region

On a sample midterm for my Calc 3 class the following question appears:

Find the mass of (and sketch) the region EE with density ρ=ky\rho = ky bounded by the ‘cylinder’ y=sinxy =\sin x and the planes z=1−y,z=0,x=0z=1-y, z=0, x=0 for 0≤x≤π/20\le x\le\pi/2.

Solution attempt: By projecting down to the x−yx-y plane the region is given by (x,y,z):0≤x≤π/2,sinx≤y≤1,0≤z≤1−y(x,y,z): 0\le x\le\pi/2, \sin x\le y\le1, 0\le z\le1-y.

So the mass of the region is ∫π/20∫1sinx∫1−y0kydzdydx\int_{0}^{\pi/2} \int_{\sin x}^1 \int_{0}^{1-y} ky dzdydx.

Performing all the iterated integrations gives k(29−π24)k(\frac{2}{9} -\frac{\pi}{24}) but the stated answer is k(16−π72)k(\frac{16-\pi}{72}).

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