Question

Find the volume of the paraboloid `x^2+y^2=4z` cut off by the plane 𝒛=𝟒

Answer 1

Paraboloid : `x^2+y^2=4z` Plane : 𝒛=𝟒

Cartesian coordinate → cylindrical coordinates

(𝒙,𝒚,𝒛) → (𝒓,𝜽,𝒛)

Put 𝒙=𝒓𝒄𝒐𝒔 𝜽 ,𝒚=𝒓𝒔𝒊𝒏 𝜽 ,𝒛=𝒛  `therefore x^2+y^2=r^2`

∴ Paraboloid : r² =4x and Plane : z = 4

If we are passing one arrow parallel to z axis from –ve to +ve we will get limits of z

`therefore r^2/4`≤ 𝒛 ≤ 𝟒
𝟎 ≤ 𝒓 ≤ 4

0 ≤ 𝜽 ≤ `pi/2`

Volume of given paraboloid cut off by the plane is given by ,

`V = 4int_0^(pi/2) int_0^4 int_(r^2/4)^4rdrd theta dz`

` = 4int_0^(pi/2) int_0^4 [4r-r^4/16]_(r^2/4)^4drd theta`

` = 4int_0^(pi/2) int_0^4 [4r-r^3/4]drd theta`

`=4int_0^(pi/2)[2r^2-r^4/16]_0^4d theta`

`=4int_0^(pi/2)[32-16]d theta`
𝑽 =𝟑𝟐 𝝅 cubic units