Question

The finite region bounded by curves y = x2 and y = 4 is rotated about the line y = −2. Sketch the region. Find the volume of the resulting solid.

Answer 1

You can use shells

Answer 2

I have the drawing

Answer 3

oh so I should use

2pi(xi)f(xi)

Answer 4

2pi(2sqrt(yi)(2+yi)delta-y

Answer 5

how do I find the radius?

Answer 6

this site is working terribly for me how did you find the radius

Answer 7

and why 2(y)^(1/2)

Answer 8

x=y^2 so y=sqrt(x)
and you need 2*sqrt(x) for the height of a shell

Answer 9

I still don't understand can anyone go over the method??

Answer 10

chinhodado is correct

the shell method essentially sums up all the surface areas of the solid, moving from inner side to outer side
since we are rotating around a horizontal axis, the integration will be with respect to y because the solid is rotating vertically
the center is at y=-2, the radius starts at 2 (distance from y=-2 to origin) then increases until y=4 and radius is 6 , thus radius = y+2
the height is the horizontal distance between 2 points on parabola in terms of y
which equals 2x or 2sqrty