Linear Algebra: Solid Enclosed

1. Oct 22, 2012

Stan12

1. The problem statement, all variables and given/known data
Let E be the solid enclosed by the paraboloid z = x2 + y2 and the plane z = 9. Suppose the density of this solid at any point (x,y,z) is given by f(x,y,z) = x2.

2. Relevant equations
x2 + y2 = r2 = 9; r = 3
∫∫∫E x2

3. The attempt at a solution
The limit of z is given z=9, found r = 3, and θ=2∏

x = rcosθ
y = rsinθ
z = z

∫∫∫ r2cos2θ r dzdrdθ

I got 729∏ / 4 as final answer

Last edited: Oct 22, 2012
2. Oct 22, 2012

LCKurtz

I don't think that is correct. Show us your limits and your work.

3. Oct 22, 2012

Stan12

z = x2 + y2 and plane z = 9

I set x2 + y2 = r2
and found that 0<r<3
and 0<z<9 since it's given
and 0<θ<2∏

x=rcosθ in cylindrical coord.

f(x,y,z) = x2 --> x2 in cylindrical coord. = (rcosθ)2

I set the function under a triple integral with restriction above

so I got ∫∫∫ (rcos)2 rdrdzdθ

4. Oct 22, 2012

Stan12

∫∫z r3 cos2θ drdθ

9 ∫ r4/4 cos2θ dθ
(9 * 34)/4 ∫cos2θ dθ
(9 * 34)/4 ∫(cos2θ + 1)/2 dθ = (9 * 34)/4 *1/2 [ (sin2θ)/2 + θ ] 0<θ<2∏

I got (9 * 34 * 2∏)/8 = 729∏/4

5. Oct 23, 2012

LCKurtz

Those are not the correct limits. If you are integrating dr first, the value of r depends on z. Your limits describe a cylinder of radius 3 and height 9, which is not what you have. The side of your object is a paraboloid. r goes from r=0 to the r on the paraboloid.

6. Oct 23, 2012

Stan12

So the limits to r is from 0 to r = √x2 + y2 or r = √z
and limit of z is from 0 to 9 ? since the paraboloid begins at 0 and enclosed by the plane at z = 9

∫∫∫ r2cos2θ rdrdzdθ

7. Oct 23, 2012

LCKurtz

Yes.

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