Question

The region R bounded by y=e^-x and y=0 and lying to the right x=0 is rotated about the y-axis

Answer 1

if you want the volume, that would be, using shells,

∫[0,∞] 2πrh dx
where r=x and h=y=e^-x

∫[0,∞] 2πxe^-x dx
= -2π(x+1)e^-x [0,∞] = 2π

using discs, it is

∫[0,1] πr^2 dy
where r = x = -ln(y)

∫[0,1] π(lny)^2 dy
= πy(2 - 2lny + (lny)^2)[0,1]
= 2π