Double Integrals with Polar Coordinates

[InertialRef]

Homework Statement

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

Homework Equations

x² + y² = r²
x = rcosθ
y = rsinθ

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! :)

 

[mfb]

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

 

[InertialRef]

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