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Now let's consider how we would handle a volume integral in
cylindrical coordinates. Let's say that I have a charged density
of four coulombs per meter cubed and I wanted to know the total
charge inside this region. Then what I would do is a triple
integral. Where I would put my function 4 meters coulomb cubed,
and I would put my DV term, which is RDRD phi DZ and I would
integrate this over the specific range. R would go from 0 to 1.
See right here is my radius. 0 to 1. Phi would go from phi
equals 0 to phi equals pi by 4. And my Z term would go from this
value right there, which is 2 to this value which is 5. And I
would do this integral. Notice that the R when we integrate that,
we're going to get R squared over 2 from 0 to 1. Phi is just
going to be phi from 0 to 4, and Z will be Z from 2 to 5. Now,
what would have happened if instead of having a constant value I
had 4 that was changing with R, which means the charged density
was much stronger out here on the outside of my little cylinder
than it was on the inside. In that case I would simply have put
the R in here and when I integrated, I would have gotten cubed
over 3 and that would have just changed the numbers that I
received somewhat. The process would still always be the same.
What are the things that I did? I found my function. I simply
plugged it in with my DV value. There's the DV term. Then I
figured out what my limits were, plugged it all in, integrated it
and came up with a value for the total volume that I was
interested in.