Finding the volume with a triple integral

[Jim4592]

Homework Statement

Use spherical coordinates to find the volume of the solid that lies above the cone z² = x² + y² and below the sphere x² + y² + z² = z

Homework Equations

I'm going to use { as an integral sign.

Volume = {{{ P² Sin[Φ] dP dΦ dΘ

The Attempt at a Solution

P² = x²+y²+z²

P² = z
P = Sqrt[z]

(P Cos[Φ])² = P² Sin²[Φ]
Cos²[Φ] = Sin²[Φ]
Tan²[Φ] = 1
Φ = Pi/4

And for the limits of integration i got:

from 0 to 2 Pi on the first integral bracket
from 0 to Pi/4 on the second integral bracket
from 0 to Sqrt[z] on the thrid integral bracket

I was hoping someone could please confirm that these are the right limits of integration before i evaluate it, thanks in advanced!

 

[vela]

The angular limits are okay, but your upper limit for ρ isn't. Try completing the square in the equation for the sphere to get it into standard form, so you have an idea of where the sphere is centered and what its radius is.

Spherical coordinates might not be the best choice here.

Edit: I see the problem says to use spherical coordinates. Oh well.

 

[Jim4592]

how do i get the equation of the sphere into standard form since the Right hand side of the equation equals z?

 

[vela]

Move the z to the LHS and then complete the square.

 

[gabbagabbahey]

Does your original problem statement explicitly tell you to use spherical coordinates centered at the origin? If not, you may find it easier to use spherical coordinates centered at (0,0,1/2)

 

[Jim4592]

Move the z to the LHS and then complete the square.

when i move it to the LHS i get: x²+y²+z²-z =0

I'm not sure how to complete the square on that. and once i have the completed square formula how do i use that to get the limits of integration?

gabbagabbahey << yes sorry it has to be at the origin.

 

[gabbagabbahey]

Just complete the square on z^2-z...

 

[Jim4592]

this completing the square is really screwing with me ha,

z^2 - z = 0

z^2 - z - 1/2 = -1/2

(z-1/2)^2 = -1/2

is that right?

 

[gabbagabbahey]

\left(z-\frac{1}{2}\right)^2=z^2-z+\frac{1}{4}

Studying university level mathematics is no excuse for forgetting basic high school algebra.

 

[vela]

Nope. If you multiply the LHS out, you get z^2-z+1/4, not what you have on the second line. Take the coefficient of z, divide by two, and add the square of that number.

 

[Jim4592]

\left(z-\frac{1}{2}\right)^2=z^2-z+\frac{1}{4}

Studying university level mathematics is no excuse for forgetting basic high school algebra.

I honestly have no idea how to complete the square then...

what I'm trying to complete the square on is z^2 -z ...right?

so to do that from the example: (the example I'm folowing is ax^2 + bx + c = 0)

step 1 says to move c to the other side: well there's no c so: z^2 - z = 0

step 2) if a does not = 1 divide by a, well a in this case = 1 so: no change

step 3) divide the z term coefficient by 2 then square it so that value is: (1/2)^2

step 4) add the term in step 3 to both sides: z^2 - z + (1/2)^2 = (1/2)^2

step 5) this is where i get confused... it just says to re-write it as a perfect square but i clearly don't know how to do it because what i tried to do came out as (z-1/2)^2 which is wrong.

I was hoping by writing this all out you can help point out which part I'm doing wrong.

 

[gabbagabbahey]

I honestly have no idea how to complete the square then...

what I'm trying to complete the square on is z^2 -z ...right?

so to do that from the example: (the example I'm folowing is ax^2 + bx + c = 0)

step 1 says to move c to the other side: well there's no c so: z^2 - z = 0

step 2) if a does not = 1 divide by a, well a in this case = 1 so: no change

step 3) divide the z term coefficient by 2 then square it so that value is: (1/2)^2

step 4) add the term in step 3 to both sides: z^2 - z + (1/2)^2 = (1/2)^2

step 5) this is where i get confused... it just says to re-write it as a perfect square but i clearly don't know how to do it because what i tried to do came out as (z-1/2)^2 which is wrong.

I was hoping by writing this all out you can help point out which part I'm doing wrong.

There is nothing wrong here. The problem is that in your previous attempt you subtracted 1/2 rather than adding (1/2)^2.

The x^2+y^2 on the LHS changes nothing from this process:

x^2+y^2+z^2-z=0 \implies x^2+y^2+\left(z-\frac{1}{2}\right)^2=\frac{1}{4}