Finding The Volume Of a Shape That Is Given By the Formula 3x^2+2y^2+z^2 leq 6.

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2022-08-17

Finding The Volume Of a Shape That Is Given By the Formula 3 x 2 + 2 y 2 + z 2 6
How can I find the volume of the shape that is given by the formula
3 x 2 + 2 y 2 + z 2 6

Answer & Explanation

Siena Bennett

Siena Bennett

Beginner2022-08-18Added 17 answers

Step 1
Dividing both sides by 6, this is an ellipsoid in standard form with semi-axes of length 6 , 3 , and 2 . It is well-known that the volume of an ellipsoid E : x 2 a 2 + c 2 b 2 + z 2 c 2 = 1
has volume 4 3 π a b c, so in this case your answer is 6 .
Step 2
If you are interested in actually carrying out the integration, then consider rewriting the equation in the above form and then performing the following change of coordinates:
{ x = 2 r cos θ sin ϕ y = 3 r sin θ sin ϕ z = 6 cos ϕ
The new bounds of integration will take r from 0 to 1, θ from 0 to 2 π, and ϕ from 0 to π. The resulting Jacobian from the change of variables will be a b c = 6. The remaining steps are pretty straightforward, so I'll leave that to you.

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