Answer to Question #143567 in Calculus for Promise Omiponle

Question

Evaluate the triple integral ∭E x dV where E is the solid bounded by the paraboloid x=5y^2+5z^2 and x=5.

Answer 1

We’ll integrate in the order dxdydz. The plane x = 5 cuts the paraboloid off in a “bowl”

shape, and so the yz bounds are the disc centered at the origin bounded by the circle of intersection of

the paraboloid and the plane x = 5, projected to the yz plane. This intersection is the circle

y² + z² = 1 (set x = 5 in the equation x = 5y² + 5z²).

$\iiint_E xdV=\iint_{y^2+z^2=1}\intop^5_{5y^2+5z^2} x dxdydz=\iint_{y^2+z^2=1} 1/2(25-25(y^2+z^2)^2)dydz=25/2\intop^{2π}_0\intop^1_0 (1-r^4)rdrd\theta$

Changing to polar for the yz plane

$=25π(1/2 -1/6) = 25π/3$

Answer: 25π/3