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## Question

My calculus teacher gave me this as a challenge problem and I enjoyed solving it so I thought I would share it.

The problem is to integrate the following function over the surface of a sphere of radius 1: |x|/(|x|+|y|+|z|).

but here is the trick: try doing it without using calculus!

I actually recommend trying it with calculus first though.

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Seems like one doogie of an integral, I'll give it a try later today if I get bored in class...

To get rid of the abs function calculate the integral only on the part of the circle where x y and z are positive then multiply by 8...

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Ok tried it but got stumped, I could code a program that would calculate the integral the hard way but I'm guessing that's not the answer you're looking for...

Does it have anything to do with the x+y+z=1 plane?

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I haven't done a surface integral in ages, so I'm probably entirely wrong here, but can we...

... solve it by using symmetry?

Integral of |x|/(|x| + |y| + |z|) over the sphere = integral of |y| / (|x| + |y| + |z|) = integral of |z| / (|x| + |y| + |z|) over the sphere.

Call this surface integral's value I.

Then 3*I = sum of integrals. Here, I think the sum of those integrals equals the integral of the sum of the integrands. So 3*I = surface integral of (|x| + |y| +|z|)/(|x| + |y| + |z|) over the sphere. That's just 1, and integrating 1 over the sphere's surface should just give us its surface area (I think).

So... 3*I = surface area of a sphere of radius 1 = 4*pi

So the answer is (4/3) pi.

This is probably based on faulty assumptions, but it really doesn't require much in the way of calculus - just algebra and a very basic sense of what integrals represent.

Edited by WitchOfDoubt
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Here's a non-calculus guess

2/3

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I haven't done a surface integral in ages, so I'm probably entirely wrong here, but can we...

... solve it by using symmetry?

Integral of |x|/(|x| + |y| + |z|) over the sphere = integral of |y| / (|x| + |y| + |z|) = integral of |z| / (|x| + |y| + |z|) over the sphere.

Call this surface integral's value I.

Then 3*I = sum of integrals. Here, I think the sum of those integrals equals the integral of the sum of the integrands. So 3*I = surface integral of (|x| + |y| +|z|)/(|x| + |y| + |z|) over the sphere. That's just 1, and integrating 1 over the sphere's surface should just give us its surface area (I think).

So... 3*I = surface area of a sphere of radius 1 = 4*pi

So the answer is (4/3) pi.

This is probably based on faulty assumptions, but it really doesn't require much in the way of calculus - just algebra and a very basic sense of what integrals represent.

WELL DONE! You got it exactly right!

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