How Do You Calculate the Volume of a Rotating Body Around Different Axes?

[BadatPhysicsguy]

Homework Statement

The curve y=1/x and the line y=2.5-x enclose an area together. Determine the exact volume of the rotating body that is formed when this field rotates about
a) The x-axis and b) The y-axis

Homework Equations

The formula for rotation around the x-axis is pi*integrate from b to a y^2 dy..
The formula for rotation around the y-axis is pi*integrate from d to c x^2 dx.

The Attempt at a Solution

I'm going to start with a). So what I've learned is that around the x-axis, first I have to find the points from where I'm going to integrate (i.e. integrate from z to y and so on). To get these two points, I'm going to put the curve equal to the line so I know where they intersect.
2.5-x=1/x => 2.5x-x^2=1 => x^2-2.5x+1 = 0 and I get x(1) = 2 and x(2) = 0.5.

So if I'm going to find out the answer, I'll also use (1/x)^2 - (2.5-x)^2. So the total would be:

pi * integrate from 2 to 0.5 for (1/x)^2 - (2.5-x)^2 which gives me the correct answer 1.125*pi.

Okay, so how do I do it for b then?
Take a look here: http://www.wolframalpha.com/input/?i=y=1/x,+y=2.5-x
I want to calculate the volume for the rotation that is both under the purple and blue line. But how do I get that? It is obvious that the "integration limits" are from 2.5 to 0 from that graph. But how do I express it in a formula?

If I do the same as earlier, (1/x) - (2.5-x) Then doesn't that area under the purple line eliminate all of it that is under the blue and then some?

 

[6c 6f 76 65]

Think of it this way:
You'll get the volume of the figure if you first find the volume under the purple line and the remove the volume $between$ the purple line and the blue curve

 

[BadatPhysicsguy]

Think of it this way:
You'll get the volume of the figure if you first find the volume under the purple line and the remove the volume $between$ the purple line and the blue curve

So first I'd take the total volume under the purple line. But how do I get the volume that is between the purple line and the blue curve? Purple minus blue and then the total volume under purple minus whatever I got from the second one?

 

[6c 6f 76 65]

So first I'd take the total volume under the purple line. But how do I get the volume that is between the purple line and the blue curve? Purple minus blue and then the total volume under purple minus whatever I got from the second one?

Exactly!

 

[SammyS]

Homework Equations

The formula for rotation around the x-axis is pi*integrate from b to a y^2 dy..
The formula for rotation around the y-axis is pi*integrate from d to c x^2 dx.

In this and your other recent thread, you have the form of the integral wrong.

For the rotation about the x-axis, y is a function of x, and the integration variable is x, not y, so that should be dx.

The integral is: $\displaystyle\ \pi\int_{x_1}^{x_2} y^2\, dx$