1. Volume of a solid

Hi, how do I solve this question?

Let R be the region that is bounded by the triangle with the vertices (0,0), (3, 0), and (1, 1).

Find the volume of the solid generated by revolving the region R about the line x = 3.

Thanks.

2. Originally Posted by coldfire
Hi, how do I solve this question?

Let R be the region that is bounded by the triangle with the vertices (0,0), (3, 0), and (1, 1).

Find the volume of the solid generated by revolving the region R about the line x = 3.

Thanks.
sketch a diagram ...

region is bounded by the lines $\displaystyle y=x$ , $\displaystyle y = 0$ , and $\displaystyle y = \frac{3-x}{2}$

rotating about the line x = 3, you can use cylindrical shells w/r to x (will require two integral expressions) or you can use disks w/r to y.

3. If I use disks, would the integral be:

$\displaystyle \pi\int_{0}^{1}(3-2y)^2-y^2 dy$

?

4. Hi codfire
Originally Posted by coldfire
If I use disks, would the integral be:

$\displaystyle \pi\int_{0}^{1}(3-2y)^2-y^2 dy$

?
You need to consider the axis of rotation. It's rotated about x = 3, not x = 0.

The formula : $\displaystyle V=\pi \int^a_b (x-k)^2~dy$ , where k is the axis of rotation.

And you also need to see which curves cover the larger volume after rotated, is it $\displaystyle y=x$ or $\displaystyle y = \frac{3-x}{2}$ ?