If someone could guide me through the process of solving this integration problem that would be very helpful.

Question

anonymous · Answer

It's an applications problem dealing with the volume generated by rotating a curve by an axis. I'll go strait to the problem: I have to rotate the region bounded by the graphs y=secx, y=1, x=-1, and x=1 with reference to the x-axis.

anonymous · Answer

The integral I came with is:$$\int\limits_{-1}^{1}((1+\sec(x))^2-1)dx$$ Before proceeding is the integrand correct?

turingtest · Answer

always graph it first...

anonymous · Answer

Thanks for the advice but I think I already graphed it and I'm struggling with the integral the graph suggests to be the answer for this problem. I don't think we have covered the indefinite integral of sec(x) in our class.

turingtest · Answer

ok, I just have to do it myself to make sure you got the right setup
I can't just "see" the answer lol, still human...|dw:1333607809980:dw|

experimentx · Answer

http://blmath.wordpress.com/2008/12/09/integral-of-cscx-and-secx/

turingtest · Answer

the way I see it the outer radius of each disk is $\sec x$ and the inner is $y=1$
going about x this gives the integral$$\pi\int \sec^2x-1dx$$so I think you had it wrong...

turingtest · Answer

this integral really should not be a problem; you should know the integral of sec^2xdx

turingtest · Answer

bounds are -1 to 1 of course...

anonymous · Answer

Did you take into account the distance between secx and y=0?

turingtest · Answer

yes, that is the outer radius$$r_o=y=\sec x$$

turingtest · Answer

the inner radius is$$r_i=y=1$$

anonymous · Answer

I think I see how you approach this, I saw the outer disk as secx+1 and the inner disk as 1. I mean their radii.

turingtest · Answer

you set up like you were revolving around the line y=-1

turingtest · Answer

ok, not quite, but almost like that
you sort of mixed the two ideas...

anonymous · Answer

Yeah I see what you mean, I was thrown off by the graph of secx and y=1 so close together.

anonymous · Answer

I really appreciate your help and time. Thanks for everything

turingtest · Answer

which is why I always say to draw the graph
do it by hand, then check with wolfram or a graphing calculator to be sure when you can
visualization is the key to these problems

anonymous · Answer

One last question though, wouldn't the inner disk be a cylinder?

turingtest · Answer

yes, here it is with a few rings drawn in|dw:1333608686746:dw|