Calculate the volume of a cube having edge length a by integrating in spherical coordinates.

onetreehillyg

onetreehillyg

Open question

2022-08-31

Finding the volume of a cube using spherical coordinates
Calculate the volume of a cube having edge length a by integrating in spherical coordinates. Suppose that the cube have all the edges on the positive semi-axis. Let us divide it by the plane passing through the points (0;0;0),(0;0;a),(a;a;0).
We get two equivalent prism; now we divide the section again by the plane passing through the points (a;0;a),(0;0;0),(a;a;a); So one may write:
1 2 V = ( 0 π / 4 d ϕ 0 π / 4 d θ 0 a cos ϕ r 2 sin ϕ d r + π / 4 π / 2 d ϕ 0 π / 4 d θ 0 a sin ϕ cos θ r 2 sin ϕ d r )
So one should expect V = a 3 but the latter expression gives a different result. What is the correct way to calculate V and why doesn't my reasoning work?

Answer & Explanation

Alison Mcgrath

Alison Mcgrath

Beginner2022-09-01Added 9 answers

Step 1
Your limits of the integration for ϕ are wrong, because for a cube ϕ = ϕ ( θ ) in your notation. The integrals with correct limits are
1 2 V = 0 π / 4 0 arctan ( a cos θ ) 0 a cos ϕ r 2 sin ϕ d r d ϕ d θ + π / 4 π / 2 arctan ( a cos θ ) π / 2 0 a sin ϕ cos θ r 2 sin ϕ d r d ϕ d θ = a 3 6 + a 3 3 = a 3 2
Step 2
Solving the integrals is a bit tedious, but straightforward. For doing the integration it is helpful to note that ( 1 cos ( ϕ ) ) 2 d ϕ = tan ϕ + C and ( 1 cos ( ϕ ) ) 3 sin ϕ d ϕ = ( 1 cos ( ϕ ) ) 2 tan ϕ d ϕ = 1 2 ( tan ϕ ) 2 + C

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