Find the volumes of the solids generated by revovling the region in the first quadrant bounded by the curve x=4y-4y^3 and the y-axis about the given axes:

Question

a. the x-axis
b. the line y=1

NO idea how to set it up!!! HElP!!

Answer 1

the function is

x = 4y(1-y^2)

so you know it crosses the y-axis at -1,0,1

So you're dealing with the bump in QI from y=0 to 1

To revolve that about the x-axis, I'd suggest shells, so

v = integral[0,1] 2pi r*h dy
where r = y, and h = x

v = 2pi integral[0,1] y(4y-y^3) dy
= 2pi Integral 4y^2 - y^4 dy
= 2pi (4/3 y^3 - 1/5 y^5) [0,1]
= 2pi (4/3 - 1/5)
= 34pi/5

Around y=1, same formula, but now r = 1-y.

Answer 2

I did that already and got it incorrect

Answer 3

nevermind i see what i did wrong