Question

How to find the volume of the region bounded by the curves `z1=x^2+y^2 and z2=1−x^2−y^2` ?

Hello, I am struggling with this question. I would be thankful if you could help me solving it.

Find the volume of the region bounded by the curves `z1=x^2+y^2 and z2=1−x^2−y^2`
Hint: To find this, you need to make a triple integral

Answer 1

`V = pi/4`

Explanation:

This is best done in cylindrical because what you have, with rotational symmetry about the y- axis, is:

• `z_1 = r^2`

• `z_2 = 1 - r^2`

The cylindrical volume element is:

• `dV= r \ dr \ d varphi \ d z`

So:

`V= int_(varphi = 0)^(2 pi) \ int_(r = 0)^(1/sqrt2) \ int_(z = r^2)^(1 - r^2) \ r \ dz \ dr \ d varphi`

`= 2 pi \ int_(r = 0)^(1/sqrt2) \ (rz)_(z = r^2)^(1 - r^2) \ dr \`

`= 2 pi \ int_(r = 0)^(1/sqrt2) \ r - 2 r^3 \ dr \`

`= 2 pi \ ((r^2- r^4)/2 )_0^(1/sqrt2)`

`= pi/4`