# Find the volume of the region bounded by y=sqrt(z-x^2) and x^2+y^2+2z=12?

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find the volume of the region bounded by y=sqrt(z-x^2) and x^2+y^2+2z=12?

find the volume of the region bounded by y=sqrt(z-x^2) and x^2+y^2+2z=12?

A graphic contribution.

A graphic contribution.

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this is symmetric about z axis so we use polar co-ords

first paraboloid

second

(and

here is a 2-D plot:

We need to find the volume that is made by revolving the area between the red and blue curves around the z-axis. That is the area under the purple curve, which is entered as

Ie, we need to find volume

where volume element in polar is

As an aside, @abubakar wanted this expressed as a triple integral . I originally skipped that step because it just adds extra notation but if we start with the general volume element and switching straight to cylindrical co-ordinates, we can say that

now neither r or z depend upon

I did that to simplify the next bit, because we have

In general terms, to do this we need either do

OR

The explanation and drawings here are helpful

The key to this is that these are iterated integrals. The

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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