Please help- area problem

whocares27 · #1 $Old$ November 29th, 2008, 11:00 AM

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

galactus · #2 $Old$ November 29th, 2008, 12:01 PM

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

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

whocares27 · #3 $Old$ November 29th, 2008, 12:52 PM

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

galactus · #4 $Old$ November 29th, 2008, 01:49 PM

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.