Setting Up Integrals: Step-by-Step Examples

[Barley]

Can anyone reccomend a website, or text, where there are step by step examples of setting up area and volume integrals where your looking for forces. I need to be able to do set ups for situations similar to the following :find the gravitational field vector on surface point of a cylinder or find or gravational force on a point mass from a disk.

Really get confused with this-- especially when it comes to putting the pieces in terms of each other?
Seems my calc. book just has me evalulate integrals and there's a big leap from my freshman/softmore physics text, where I hardly did any calc., to my junior level text.


Thanks.

 

[Astronuc]

See if these help.

http://mathworld.wolfram.com/Integral.html

http://mathworld.wolfram.com/SurfaceIntegral.html

http://www.maths.abdn.ac.uk/~igc/tch/ma1002/int/int.html

http://library.thinkquest.org/3616/Calc/S3/S3.html - Integral Calculus (basic) with examples of area and volume integrals

http://en.wikipedia.org/wiki/Integral_calculus (pretty basic)

If not, we'll try again.

 

[Barley]

Peliminary inspection of these sites shows a lot about techniques for solving integrals but not a lot on setting one up for the situations I tried to describe. Maybe what I'm looking for is a good mechanics text. The one I have has examples in it that I can't follow, not because my integration is rusty, but because my text starts with a simple enough relation and the next line is the result of a triple integration.

Example of Hw problem:

Calculate the gravitational field vector due to a homogeneous cylinder at exterior points on the axis of the cylinder.
Only because the problem states that the result is to be found by computing the force directly; start with g = F/m

g = -GMrhat/ r^2


I can get that symmetry gives us that there is only force in z direction, and choosing a reference point on the z axis and pick an arbitrary point on the surface of mass dm- where dm = rhodV . The point dm connects to the (0, 0, z) reference point with a radial line and makes an angle with the z axis that we can put into the integral as the magnitude of dgz, so what goes into the integral is cos(angle)--where cos(angle) =(zo-z)^2/((sqrt r^2 + (zo -z)^2)).

Now, there's rhodV = dm = rhodr rdangle dz

Somehow, I have in my notes the final integral, skipping the 3 limits of integration resloves itself into, bringing rho outside, rho///drdangledz(zo -z)/(((zo -z)^2 + r^2))^3/2)). Even if there's an error in my notes, I'm stuck on the set up.

I can't figure out why the top term is no longer squared. Looking back at the origional formula; gz = -Grho(integral)cosangle/r^2.
I've drawn a triangle on my picture connecting the z axis across to the surface point, to the radial line, and back to the point zo. Now, I'm confused-- I've labled the radius of the cylinder R and the radial vector from reference point to zo is labled little r. To evaluate the integrand I need to get r in terms of R ? Just stuck!

See, what I need are some examples with some intermediate steps in setting up these types of problems.

Thanks

 

[Barley]

Solved it

ok it was simple and I apologize to anyone who tried to read my post. The (zo -z)^2 term never belonged there- Just lack of sleep r in terms of R easy.
No biggie-- hard part, I know is evaluating the result of the set up but looks like integration by parts--
Still, need practice, and worked out examples would be of a lot of help.

Reccomendations, advice, appreciated.