Question to do with volume of a solid between a paraboloid and a plane

**rpatel** · January 26th 2010, 07:11 AM

Hi

the question is

Use polar coordinates to find the volume of the solid which is under the paraboloid $z=x^{2}+y^{2}$ and above the disk $x^{2}+y^{2}\leq9$ in the plane $z=0$ .

thanks

**shawsend** · January 26th 2010, 09:15 AM

If $z=x^2+y^2$ then $z=f(r,t)=r^2$ so we integrate over the circle $x^2+y^2=9$ as the radius goes from 0 to 3 and all the way round the circle or just 4 times the part in the first quadrant since f is symmetrical:

$V=4\int_0^{\pi/2}\int_0^3 r f(r,t) dr dt$

and you can check that by just calculating the volume inside the paraboloid via shells and then subtract that from just the cylinder it's housed in:

$\pi (3)^2 9-\pi\int_0^9 ydy=4\int_0^{\pi/2}\int_0^3 r f(r,t) dr dt$

half-in and half-out right?

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