Suppose E is the region bounded above by the cylinder x^2 + z^2 = 5, below by the plane z=1,and on the sides by the planes y=−1and y=2.?
Suppose E is the region bounded above by the cylinder x^2 + z^2 = 5, below by the plane z=1,and on the sides by the planes y=−1and y=2. Find ∫∫∫ z dv
If someone can just set it up. Thanks
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In the x-z plane (where y = 0), you have a circular segment whose (x,z) vertices are (-2,1) and (2,1), round on the top and flat on the bottom. Because your integrand is z dV, you should probably slice this area parallel to the x axis. The endpoints of each slice will be
(-sqrt(5-z^2), z) and (+sqrt(5-z^2), z). You will integrate
2z*sqrt(5 - z^2) dz
from z = 1 to z = sqrt(5).
It's a pretty easy integration. When you've finished with it, you can take care of the 3rd (y) dimension by simply multiplying your result by the quantity (2 - (-1)), that is, multiply by 3.