Finding limits of integral in spherical coordinates

[uzman1243]

Homework Statement

The question asks me to convert the following integral to spherical coordinates and to solve it

Homework Equations

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?

 

[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?

 

[uzman1243]

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?

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 I am guessing
r goes from 0 to 2
∅ and θ from 0 to 2π

 

[ehild]

Are you sure in 2pi?

 

[uzman1243]

Are you sure in 2pi?

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

 

[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?

 

[uzman1243]

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?

View attachment 82540

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

 

[ehild]

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

 

[uzman1243]

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

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

 

[BvU]

Looks good.

 

[ehild]

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

Yes.