Disc Method Finding Volume Trig Function

[Chaoticoli]

Homework Statement

Consider the region of the x-y plane between the line y=7, the curve y=3sin(x)+4 , and for
-pi/2<x<3pi/2

Find the volume of the solid generated by revolving this region about the line y=7.

I know that for this problem, I will be using the disk method (as the title states). I drew the graph and I find that there are two areas and since the graph is symmetric, the two areas are equal.

A picture of the graph is here: http://www.wolframalpha.com/input/?i=+the+curve+7=3*sin(x)+4+between+-pi/2<x<3pi/2

I know I must find the area of a slice. I am just not exactly sure how to find it. I know the equation for the area is A(x) = pi*r(x)^2

And then, you need to find the limits of integration and with those limits, the volume will be:

Volume = ∫ from a to b of pi*A(x)

Any help would be greatly appreciated ! :)

 

[Chaoticoli]

Obviously, I just need to find out what r(x) is and figure out if I should integrate with respect to x or y.

 

[haruspex]

since the graph is symmetric, the two areas are equal.

Not exactly. The area of interest is above the curve, so they're not so much equal as, in a sense, complementary.
Anyway, finding the area of regions in this graph won't help you.

I know I must find the area of a slice. I am just not exactly sure how to find it. I know the equation for the area is A(x) = pi*r(x)^2

Yes.
Can you visualise the rotation about y=7?
r(x) is the distance between two y values at x. What two y values?

 

[Chaoticoli]

Yes. I visualize it being rotated, forming discs about the line y=7.

Is it the distance between y=3sin(x)+4 and y=7?

(3sin(x) + 4) - 7 = 3sin(x) -3 = r(x) ??

 

[Chaoticoli]

Actually, I just figured it out :). I got 27pi^2 after integrating from -pi/2 to 3pi/2 of (3sin(x)-3)^2 dx