Question

Find the volume of the region enclosed by the cylinder x squared plus y squared equals 36 and the planes z equals 0 and y plus z equals 36.

Answer 1

`\mathbf{V = 1296 \pi }`

Explanation:

Given that :

Find the volume of the region enclosed by the cylinder
`x^2 + y^2 =36` and the plane z = 0 and y + z = 36

From y + z = 36

z = 36 - y

The volume of the region can be represented by the equation:

`V = \int\limits \int\limits_D(36-y)dA`

In this case;

D is the region given by
`x^2 + y^2 = 36`

Relating this to polar coordinates

x = rcosθ y = rsinθ

x² + y² = r²

x² + y² = 36

r² = 36

r =
`√(36)`

r = 6

dA = rdrdθ

r → 0 to 6

θ to 0 to 2π

Therefore:

`V = \int\limits^(2 \pi) _0 \int\limits ^6_0 (36-r sin \theta ) (rdrd \theta)`

`V = \int\limits^(2 \pi) _0 \int\limits ^6_0 (36-r^2 sin \theta ) drd \theta`

`V = \int\limits^(2 \pi) _0 [(36r^2)/(2)- (r^3)/(3)sin \theta]^6_0 \ d\theta`

`V = \int\limits^(2 \pi) _0 [648- (216)/(3)sin \theta]d\theta`

`V = \int\limits^(2 \pi) _0 [648+(216)/(3)cos \theta]d\theta`

`V = [648+(216)/(3)cos \theta]^(2 \pi)_0`

`V = [648(2 \pi -0)+(216)/(3)(1-1)]`

`V = [648(2 \pi )+(216)/(3)(0)]`

`V = 648(2 \pi )`

`\mathbf{V = 1296 \pi }`