Finding volume of sphere cartesian to polar

oliadas73 2022-10-02 Answered
Finding volume of sphere cartesian to polar
Using cartesian coordinates the volume of a sphere can be calculated via
R R R 2 z 2 R 2 z 2 R 2 z 2 y 2 R 2 z 2 y 2 1 d x d y d z
However using a parametrisation with polar coordinates the Integral becomes:
0 2 π 0 R 0 π 1 R 2 sin ( θ ) d θ d R d φ .
Thinking closer about this I just don't see how these integrals are interchangeable. Notably substituting polar coordinates in the cartesian one I don't notice how the borders simplify.
For instance using the substitution
( x y z ) = ( R cos ( φ ) sin ( θ ) R sin ( φ ) sin ( θ ) R cos ( θ ) )
I get for the first border:
R 2 ( 1 cos 2 ( θ ) sin 2 ( θ ) sin 2 ( φ ) ) . What's that even?
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Answers (2)

selixaak
Answered 2022-10-03 Author has 6 answers
Step 1
Such as x = R 2 z 2 y 2
R R R 2 z 2 R 2 z 2 R 2 z 2 y 2 R 2 z 2 y 2 1 d x d y d z
so we can also do
2 0 π 0 2 π 0 R ρ 2 s e n φ d ρ d θ d φ
with x = ρ s e n ( φ ) c o s ( θ )
y = ρ s e n ( φ ) s e n ( θ )
z = ρ c o s ( φ )
Step 2
Then x 2 + y 2 + z 2 = ρ 2 s e n 2 ( φ ) ( c o s 2 ( θ ) + s e n 2 ( θ ) ) + ρ 2 c o s 2 ( φ ) = ρ 2 s e n 2 ( φ ) + ρ 2 c o s 2 ( φ ) = ρ 2
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Lisantiom
Answered 2022-10-04 Author has 2 answers
Explanation:
If x = R cos ( φ ) sin ( θ ), y = R sin ( φ ) sin ( θ ), and z = R cos ( θ ), then
x 2 + y 2 + z 2 = R 2 cos 2 ( φ ) sin 2 ( θ ) + R 2 sin 2 ( φ ) sin 2 ( θ ) + R 2 cos 2 ( θ ) = R 2 ( sin 2 ( θ ) + cos 2 ( θ ) ) = R 2 .
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