Find the volume of a solid which is bounded by the paraboloid 4z=x^2+y^2, the cone z^2=x^2+y^2 and the cylinder x^2+y^2=2x.

vroos5p

vroos5p

Answered question

2022-08-11

Find the volume of a solid which is bounded by the paraboloid 4 z = x 2 + y 2 , the cone z 2 = x 2 + y 2 and the cylinder x 2 + y 2 = 2 x
I approached this problem by trying to find the volume bounded by the paraboloid and the cylinder and then subtracting it from the volume bounded by the cone and the cylinder. But I am getting the wrong answer. I converted the all the bounds into cylindrical co-ordinates.
For finding the volume bounded by the cone and the cylinder,
Bounds of integration:   z = 0   to   z = r     ,     r = 0   to   r = 2 cos ( θ )     ,     θ = 0   to   θ = 2 π  .
For finding the volume bounded by the paraboloid and the cylinder,
Bounds of integration:   z = 0   to   z = r 2 4     ,     r = 0   to   r = 1     ,     θ = 0   to   θ = 2 π  .

Answer & Explanation

kilinumad

kilinumad

Beginner2022-08-12Added 21 answers

Step 1
Your bounds for θ are wrong; θ should run from π 2 to π 2 :
V = π 2 π 2 0 2 cos θ r 2 4 r r d z d r d θ = 32 9 3 8 π .
Step 2
If you want to do it by subtracting the volume bouded by the paraboloid to the one bouded by the cone, then V = π 2 π 2 0 2 cos θ 0 r r d z d r d θ π 2 π 2 0 2 cos θ 0 r 2 4 r d z d r d θ = = 32 9 3 8 π .

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