Volume of Solid

Question

Find the volume of the solid below z=36 − x^2 − y^2 over the region bounded by

x^2 + y^2 = 4 and x^2+y^2=36



Use polar coordinates. Give your answer to 2 decimal places.

Answer 1

The question asks us to find the volume under a paraboloid z = 36 - x^2 - y^2 over the annulus 4 ≤ x^2 + y^2 ≤ 36.

Since we have an annulus, changing the coordinate system to cylindrical should be our first step:

1. x = rcos(θ), y = rsin(θ)
2. x^2 + y^2 = r^2
3. z = 36 - r^2
4. dA = r dr dθ
5. Region: 2 ≤ r ≤ 6, 0 ≤ θ ≤ 2π

Now we can find the volume by integrating over the area:

V = ∬(36 - x^2 - y^2) dA

= ∫(θ=0 to 2π) ∫(r=2 to 6) (36 - r^2) * r dr dθ

Compute inner integral:

∫(r=2 to 6) (36 - r^2) * r dr

= ∫(2 to 6) (36r - r^3) dr

= [18*r^2 - (1/4)r^4] from 2 to 6

= (186^2 - (1/4)6^4) - (182^2 - (1/4)2^4)

= 324 - 68

= 256

Now integrate over θ:

V = ∫(0 to 2π) 256 dθ = 256 * 2π = 512π

512π = 512 * 3.14 = 1608.50