Linear Algebra: Solid Enclosed

[Stan12]

Homework Statement

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.

Homework Equations

x² + y² = r² = 9; r = 3
∫∫∫_E x²

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

 

[LCKurtz]

Homework Statement

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.

Homework Equations

x² + y² = r² = 9; r = 3
∫∫∫_E x²

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

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

 

[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

 

[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

 

[LCKurtz]

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

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

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.

 

[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

 

[LCKurtz]

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

Yes.