Guest Posted November 23, 2011 Report Share Posted November 23, 2011 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. Quote Link to comment Share on other sites More sharing options...

0 Guest Posted November 24, 2011 Report Share Posted November 24, 2011 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... Quote Link to comment Share on other sites More sharing options...

0 Guest Posted November 25, 2011 Report Share Posted November 25, 2011 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? Quote Link to comment Share on other sites More sharing options...

0 WitchOfDoubt Posted November 26, 2011 Report Share Posted November 26, 2011 (edited) 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 November 26, 2011 by WitchOfDoubt Quote Link to comment Share on other sites More sharing options...

0 bushindo Posted November 26, 2011 Report Share Posted November 26, 2011 Here's a non-calculus guess 2/3 Quote Link to comment Share on other sites More sharing options...

0 Guest Posted November 27, 2011 Report Share Posted November 27, 2011 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! Quote Link to comment Share on other sites More sharing options...

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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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