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August 26th, 2008, 01:27 PM   #1
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Volume

Hi,

Let the volume D be:

x >= 0
y >= 0
x+y<= 1
0<= z <= (x-y)/2

Compute the volum of D.

Thx

J.
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August 27th, 2008, 01:18 PM   #2
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Re: Volume

Quote:
Originally Posted by Jinx2
Hi,

Let the volume D be:

x >= 0
y >= 0
x+y<= 1
0<= z <= (x-y)/2

Compute the volum of D.

Thx

J.
This can be set up as an integral calculus problem, where the main problem is setting up the integration limits for x, y, and z. The integrand itself is 1.

Step 1: z integral = (x^2-y^2)/2
Step 2: set up the x,y integral. It will be a subset of the plane bounded by the x axis, the line y=x and the arc of the unit circle from the x axis to the y=x line. The figure looks like a pizza slice, 1/8 of a pie.
Step 3: convert the integral to polar coordinates. Then the domain of integral is r from 0 to 1 and t (angle) from 0 to pi/4.

I got an answer of 1/16.
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