# Double Integrals with Polar Coordinates

1. Jul 4, 2013

### InertialRef

1. The problem statement, all variables and given/known data

Use polar coordinates to find the volume of the solid bounded by the paraboloid z = 47 - 5x2 - 5y2 and the plane z = 2.

2. Relevant equations

x2 + y2 = r2
x = rcosθ
y = rsinθ

3. The attempt at a solution

I substituted the z = 2 into the equation given,

2 = 47 - 5x2 - 5y2
45 = 5x2 + 5y2
9 = x2 + y2

So from here, I know that r = 3, and 0<r<3.
Since it's a circle, I know that 0<θ<2∏

Then, I know that x2 + y2 = r2, so,

z = 47 - 5(x2+y2)
= 47 - 5(r2)

∫[0,2∏]∫[0,3] (47 - 5(r2))rdrdθ

When I take the double integral, I get (441∏)/2. This is incorrect. It seems like a very simple question, and my math looks correct. Have I made a conceptual mistake somewhere?

Thanks for any help! :)

2. Jul 4, 2013

### Staff: Mentor

The height of the solid object is not the function value, as your lower border is z=2 instead of z=0.

3. Jul 5, 2013

### InertialRef

Ohhh. Yes, that makes sense. Thank you so much! :)