Triple integral over a sphere in rectangular coordinates

Batmaniac
Messages
22
Reaction score
0

Homework Statement



Evaluate the following integral:

<br /> \iiint \,x\,y\,z\,dV<br />

Where the boundaries are given by a sphere in the first octant with radius 2.

The question asks for this to be done using rectangular, spherical, and cylindrical coordinates.

I did this fairly easily in spherical and rectangular coordinates, except for the fact that I got two different answers and I can't figure out where I went wrong! That's not a problem though because I can fix that.


The Attempt at a Solution



How would I do this problem in rectangular coordinates? My integral would look like this:

<br /> \int_{0}^{{\sqrt{4-x^2-y^2}}}\int_{0}^{{\sqrt{4-x^2-z^2}}}\int_{0}^{{\sqrt{4-z^2-y^2}}}xyz\,dz\,dy\,dx<br />

Which, without some clever transformations and an extremely messy Jacobian calculation, looks unsolvable.
 
Physics news on Phys.org
I think you need to rethink your bounds on that one...
 
How does this look then?

<br /> \int_{0}^{2}\int_{0}^{2}\int_{0}^{{\sqrt{4-z^2-y^2}}}xyz\,dz\,dy\,dx<br />
 
Hmm, MATLAB tells me that's zero.
 
Your dy limit should depend on x.
 
Batmaniac said:
How does this look then?

<br /> \int_{0}^{2}\int_{0}^{2}\int_{0}^{{\sqrt{4-z^2-y^2}}}xyz\,dz\,dy\,dx<br />

That would be over a square in the xy-plane rising up to the sphere.

Projecting the sphere into the xy-plane gives you the quarter circle x2+ y2= 4, with 0\le x\le 2, 0\le y\le 2. You can let x go from 0 to 2 but then, for each x, y ranges from 0 to \sqrt{4- x^2}.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
Back
Top