Question

Given 3 surfaces, how can you obtain the plot and bounds for the solid made by intersection by all 3 surfaces?
S1 is the xy plane, z=0
S2 x^2+y^2=1
S3 z=1-x^2

Answer 1

Typically, examine cross sections and understand intersections.

x^2 + y^2 = 1 is a cylinder, parallel to the z-axis and centered at (0,0) in the x-y plane. It has a radius of 1. Important points are (1,0,0);(0,1,0);(-1,0,0);(0,-1,0). Neither x, not y can exceed 1, positive or negative.

z = 1 - x^2 is parabolic in the x-z plane and runs parallel to the y-axis. A really long tent. This is necessarily less than 1 in the z-direction. Important points are (0,0,1);(1,0,0);(-1,0,0)

Give it some thought.

Answer 2

Can I use cylindrical cordinates?

knowing x^2+y^2=1 can be replaced with x=r*cos(t) and y=r*sin(t)

also we got z=1-x^2 which can be written \[z=1-r^2\cos^2(t)\]

0<z<1 and t from 0-2Pi and r from 0-1?

Answer 3

\[\int\limits \!r\,{\rm d}[z={0\ldots 1-{r}^{2} \left( \cos \left( \theta \right) \right) ^{2}},r={0\ldots 1},\theta={0\ldots 2\,\pi}]\]

Still cant plot it thought

Answer 4

Please do.

Answer 5

For the integral, the upper bound on \(z\) needs to be \(1-\cos^2\theta=\sin^2\theta\). No \(r\) needed here, that only applies in the general coordinate conversion.

Answer 6

Thanks Holster, I need to go thru trig identities again it seems like.

My best attempt to plot this thing is hillarious :P

Answer 7

Looks good, though that's just the lateral surface of the solid (the part of the cylinder below the parabolic cylinder and above the plane).

Answer 8

Finally :)

Answer 9

Look at you!! Good work. Excellent practice.

One word of warning. 3D is as far as your plotting and imagery can go. One day, maybe, you will need a 4D integral. No visualization, there. It all becomes much more abstract. It is good to think about the abstract before you must.

Answer 10

Thanks. It took quite a bit of time time though xD

4D integral, didn't even know it was a thing xD what on earth are they used for?

Answer 11

Oh, there is FAR MORE mathematics out there than we can use to answer the "used for"
question.

Answer 12

One use could be finding a probability with respect to a multivariate probability distribution that depends on four random variables.

Another perhaps more nebulous application might be finding the "volume" (or more generally, measure) of a four dimensional object.