Question

Let R be the first quadrant region enclosed by the graph of y= 2e^-x and the line x=k.

a) Find the area of R in terms of k.

b) Find the volume of the solid generated when R is rotated about the x-axis in terms of k.

c) What is the volume in part (b) as k approaches infinity?

Answer 1

a) For the first quadrant region, x>0 to x = k, and the enclosed area is
Integral y dx =
Integral 2e^-x dx
x = 0 to k
= -2 e^-k + 2 e^0
= 2(1 - e^-k)

b) Make the integrand pi*y^2 dx and perform the resulting integration from 0 to k

c) This should be obvious after doing (b)