How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #y = 15e−x^2#, y = 0, x = 0, x = 1 revolved about the y-axis?
1 Answer
Explanation:
If you imagine an almost infinitesimally thin vertical line of thickness
#delta A ~~"width" xx "height" = ydeltax = f(x)deltax#
If we then rotated this infinitesimally thin vertical line about
#delta V~~ 2pi xx "radius" xx "thickness" = 2pixdeltaA=2pixf(x)deltax#
If we add up all these infinitesimally thin cylinders then we would get the precise total volume
# V=int_(x=a)^(x=b)2pixf(x) dx #
So for this problem we have:
# \ \ \ \ \ V=int_0^1 2pix(15e-x^2) dx #
# :. V=2pi int_0^1 (15ex-x^3) dx #
# :. V=2pi [15ex^2/2-x^4/4]_0^1 #
# :. V=2pi [15/2e-1/4] #