a rectangle R of length l and width w is revolved around the line L. Find volume of resulting solid of revolution.

2. The theorem of Pappus may be a cool way to go with this one.

Find the centroid of the rectangle. Rectangles are easy for that. It is in the center.

Find the distance the center of the rectangle is from what it is being revolved around. It's area is merely lw.

Volume would then be $V=\left(\text{area of rectangles}\right)\cdot\left(\text{distance traveled by centroid}\right)$

$V=2{\pi}\cdot \overline{x}\cdot (\text{area of rectangle})$

3. is there any way to do this problem without that theorem. . . that theorem is awesome though

4. You could try to find the equations of the lines making up the sides of the rectangle.

But Pappus is a decent way it would appear.

The centroid is located a distance of $d+\frac{\sqrt{l^{2}+w^{2}}}{2}$ from the vertical line L.

The area of the region is simply lw.