when do you subtract the x=constant from the shell function rather than place it as an upper or lower bounds?

Question

Working through examples I find that sometimes the right thing to do is subtract that constant and other times it hangs around as an upper or lower bound how do I know when to do what

Answer 1

The limits of integration apply to the region being revolved around the axis of revolution. Assuming you are revolving a region about the line x = c, that value c does not come into play when determining the limits of integration. Using the shell method it does come into play when coming up with an expression for the average radius (the distance from the center of a typical rectangle to the axis of revolution). This is likely what the original post was questioning (where and when does the "c" get subtracted).

Take a look at this graph to see if the explanations there make this a bit clearer:

desmos.com/calculator/fme5b6eywx

Answer 2

I'll be glad to help go over a set of questions relating to finding volumes by cylindrical shells if you need some help with problems like that.

It sounds like you are working some where the axis of rotation is a vertical line ( like x = h ) other than the y-axis, and some where a vertical line is a boundary of the region, and perhaps some that fit both of these descriptions. This can be confusing at first, but with care and time, one can make sense of all the nuances.

Text explanations can only take one so far, and the whiteboard feature we'll have access to in a lesson will allow a much better breakdown using sketches. Sketching these problems, in 2-D and 3-D, is a necessary if difficult part in learning how to set up the integrals, both the integrand and the limits of integration.

Let me know if I can help with a specific example, or if you would like me to provide a set that may cover all the cases. Good luck with studies, and know there are tutors like myself ready to help if needed. Thank you!

Answer 3

It sounds like the confusion might be due to the difference between integration variables and parameters in definite integrals. In particular, you might be using the word "subtract" when what you really mean is "remove" — as in, when should x appear inside the integrand (e.g., in the shell radius), and when should it be used as a bound?

Let’s clarify that with a surface area example.

When you’re computing the surface area of a shell from x = a to x = b, where a and b are constants, and the radius of that shell is a function r(x), the formula is:

S = integral from a to b of 2π*r(x)dx

However, if you want to see how that area changes as the solid expands or contracts — that is, you want to define a function S(x) — then the upper limit of integration becomes variable. In this case, we introduce a dummy variable (say t) to separate the integration from the outer variable:

S(x) = integral from a to x of 2π*r(t)dt

Here, we replaced r(x) with r(t) and dx with dt. That’s because the integration is with respect to t, while x is treated as a parameter — the limit of integration. After evaluating the integral, the result is a function of x.

In summary:

1. Expressions like x = 2 typically appear as bounds when defining the region being measured.
2. When both bounds are constants, x does not appear in the integral limits at all; but is used in the integrand.
3. When the size of the region is variable, x appears as a limit of integration, and the integrand must use a dummy variable instead (commonly t) to avoid confusion.

I hope this answer helps, and please let me know if you have any further questions!