Find the volume of this set. Let Omega={(x,y,z):x^2+y^2 leq z^2 leq 4}

crazygbyo

crazygbyo

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2022-08-18

Computing volume of a set in R 3
Let Ω = { ( x , y , z ) : x 2 + y 2 z 2 4 }.
So, the problem is to find the volume of this set. MY though: comput Ω d V
But, im having trouble seeing the picture of the set, and hence finding limit of integration. Can someone help me?

Answer & Explanation

afinat4s

afinat4s

Beginner2022-08-19Added 13 answers

Step 1
Another way to compute this is using integration. Since Ω = { ( x , y , z ) | x 2 + y 2 z 2 4 }.
Consider the upper cone (in the sense of z 0. Then, on the upper cone, x 2 + y 2 z 2, 4 x 2 y 4 x 2 , 2 x 2
Ω d V = 2 2 2 4 x 2 4 x 2 x 2 + y 2 2 1   d z d y d x
Step 2
which upon changing to cylindrical coordinates is indeed
Ω d V = 2 0 2 π 0 2 r 2 r d z d r d θ = 16 π 3
Janiya Rose

Janiya Rose

Beginner2022-08-20Added 2 answers

Explanation:
It's a double cone with an axis Z of height 2 and with a radius of the base 2. Hence V = 2 π r 2 h 3 = 2 π 4 2 3 = 16 3 π

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