# Homework Help: Revolution about y=-1

1. Sep 13, 2012

### iRaid

1. The problem statement, all variables and given/known data
Find the volume of the solid obtained by rotating the region
bounded by the given curves about the specified line. Sketch the
region, the solid, and a typical disk or washer.

y=sinx, y=cosx, 0≤ x≤∏/4, About y=-1

2. Relevant equations

3. The attempt at a solution
Tried 2 ways, shell method and washer method..

$$\pi \int_0^\frac{\pi}{4}(cosx-sinx)^{2}\,dx - \pi \int_0^\frac{\pi}{4}(-1)^{2}\,dx$$
$$\pi \int_0^\frac{\pi}{4}cos^{2}x-2sinxcosx+sin^{2}x\,dx - \pi \int_0^\frac{\pi}{4}1\,dx$$
^Hard integral

$$2\pi \int_0^\frac{\pi}{4}(x+1)(cosx-sinx)\,dx$$
$$2\pi \int_0^\frac{\pi}{4}xcosx+cosx-xsinx-sinx\,dx$$
$$2\pi (sinx+cosx)|$$ (From 0 to ∏/4, not sure how to do this in latex)
(And so on)

Thanks for any help.

Last edited: Sep 13, 2012
2. Sep 13, 2012

### daveb

You might want to relook at the way you set up the problem.

Your first integral is some finite value, let's say 1 (I know that isn't right, but follow the idea). Now imagine, instead of the line y = -1, it is rotated about y = -100. Your volume should be bigger, right? But the way you have the equation, it's getting smaller.

So, for starters, imagine instead you rotate about the line y = 0. What would each individual part look like (the Cos part minus the sin part). Now shift the radius by 1 larger (since you are rotating about y=-1). What is the radius of the outer diameter as you go from 0 to pi/4?

3. Sep 13, 2012

### SammyS

Staff Emeritus
The result is the same as subtracting,
the volume generated by rotating the region bounded by y=sin(x), 0≤ x≤∏/4, about y=-1
FROM
the volume generated by rotating the region bounded by y=cos(x), 0≤ x≤∏/4, about y=-1 .

4. Sep 13, 2012

### iRaid

@SammyS, OK, but it's still a hard integral.

Last edited: Sep 13, 2012
5. Sep 13, 2012

### iRaid

Ah I see, I thought it was becoming a cylinder..

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