Finding limits of integral in spherical coordinates

1. Apr 25, 2015

uzman1243

1. The problem statement, all variables and given/known data
The question asks me to convert the following integral to spherical coordinates and to solve it

2. Relevant equations

3. The attempt at a solution
just the notations θ = theta and ∅= phi

dx dy dz = r2 sinθ dr dθ d∅
r2 sinθ being the jacobian

and eventually solving gets me
∫ ∫ ∫ r4 *sin2θ * sin∅ dr dθ d∅

How do I find the limits now?

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2. Apr 25, 2015

ehild

Use the limits in the Cartesian system to figure out the enclosed shape. What is the minimum and maximum value of z? Those of y and x?

3. Apr 25, 2015

uzman1243

x goes from -2 to 2
y goes from 0 to √4-x2 circle with radius 2
z from 0 to √4-x2-y2 sphere with with radius 2

so Im guessing
r goes from 0 to 2
∅ and θ from 0 to 2π

4. Apr 25, 2015

ehild

Are you sure in 2pi?

5. Apr 25, 2015

uzman1243

ok so ∅ goes from 0 to 2pi as it is some sort of sphere/ elipse. correct?

6. Apr 25, 2015

ehild

See picture. Yes, the shape is spherical, but you have to integrate with respect to y from zero to some positive value, goes it round a whole circle?

7. Apr 25, 2015

uzman1243

ahhh so θ goes from 0 to pi
∅ goes from 0 to 2pi
and r goes from 0 to 2
correct?

8. Apr 25, 2015

ehild

Those were the limit for the whole sphere. But the integration does not go for negative z values, neither for negative y values.

9. Apr 26, 2015

uzman1243

r goes from 0 to 2
theta goes from 0 to pi/2
phi goes from 0 to pi
correct?

Last edited: Apr 26, 2015
10. Apr 26, 2015

BvU

Looks good.

11. Apr 26, 2015

Yes.