Let f(x,y)=1-x^2/4-y^2 and Omega={(x,y) in mathbb{R}^2:f(x,y) geq 0}. Compute the volume of the set A={(x,y,z) in mathbb{R}^3:(x,y) in Omega, 0 leq z leq f(x,y)}

Nathanael Perkins 2022-09-16 Answered
Evaluating the Volume of a Cupola-Shaped Set by Integration
Let f ( x , y ) = 1 x 2 4 y 2 and Ω = { ( x , y ) R 2 : f ( x , y ) 0 } .
Compute the volume of the set A = { ( x , y , z ) R 3 : ( x , y ) Ω , 0 z f ( x , y ) } .
My idea is to slice the set along the z-axis, obtaining a set E z - in fact, an ellipse - and computing the volume as 0 1 E z d x d y d z.
However, I am stuck finding a way to describe E z . What is the best strategy to do that?
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Answers (1)

ahem37
Answered 2022-09-17 Author has 15 answers
Step 1
Use this parameterization for the whole 3 dimensional space
x = 2 u cos v y = u sin v z = w 0 u < , 0 v < 2 π , < w <
then the integral for volume will be
V = v = 0 2 π u = 0 1 w = 0 1 u 2 ( x , y , z ) ( u , v , w ) d w d u d v
Step 2
Where ( x , y , z ) ( u , v , w ) = | x u x v x w y u y v y w z u z v z w | = | 2 cos v 2 u sin v 0 sin v u cos v 0 0 0 1 | = 2 u is the jacobian.

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