Question

The region between the graphs of x=y^2 and x=3y is rotated around the line y=3.
Find the volume

Answer 1

?

Answer 2

Ok, so you need first to draw a diagram of the region. Where are the end-points?

Answer 3

i cant tell

Answer 4

it looks something like this

Answer 5

yes exactly so the origin and what's the other pt?

Answer 6

It's when x = 3y and x = y^2. i.e., when 3y = y^2

Answer 7

9?

Answer 8

Hence x = 9; i.e., the other end point is (9,3)

Answer 9

Now you're rotating about the line y = 3, so you need to measure distance from that line. You're going to want to the integral
\[ 2\pi \int_0^9 (some \ function)^2 \ dx \]

Answer 10

You need now to figure out that function.

Answer 11

Here's one piece of it. The line x = 3y, or y = x/3, is for a given x a distance (3 - x/3) from the line y = 3.

Now figure out the distance of the other curve from the line y = 3.

The function you're squaring is
( (distance to curve #1) - (distance to curve #2) )

Answer 12

are you sure its 2 pi?