Question

Given `f(x, y)=x^2+y^2-2x`, how do you the volume of the solid bounded by `z=(f(x, y)+f(y,x))/2-5/2, z = +-3?`

Answer 1

`9pi` cubic units.

Explanation:

The section of this sold by a plane parallel to the xy-plane is the

circle with

center at `(1/2, 1/2, z)`and radius `R(z)= sqrt(z+3/2)`

For integration to find the volume V, choose an element in the form

of a circular disc of thickness `Delta z` and radius R. The faces of

this disc are parallel to the xy-plane.

Now, V = limit `Delta z to 0` of `sum piR^2 Delta z=int piR^2 dz`,

between the limits,from `z = -3` to z = 3..

So, `V = .pi int (z+3/2) dz`, between the limits

`=pi[z^2/2+3/2z],` between `z = -3 and z = 3`

`= 9pi` cubic units.