# MvCalc65

The given integral is

Where E is the region bounded by the paraboloid and the plane

First, we'll set up the limits of integration. The bounds are a paraboloid along the x-axis intersected by the y-z plane. So, the limits are:

Solve...

Now changing to polar cordinates, on the y-z plane:

and for the limits of integration r = 1 and the intersection is the unit circle on the y-z plane so:

NOTE THAT: and by double-angle formula. Divide by 2 and square both sides

So that... Plug this into integral.

Substitute u=2*θ* so 1/2du=d*θ* and when *θ* = 2π then u = 4π and when *θ* = 0 then u = 0

By half-angle formula

Factor constants and integrate term by term

Substitute v=2u so 1/2dv=du and when u = 4π then v = 8π and when u = 0 then v = 0

Hence...