Asked by Josh
The volume of the solid obtained by rotating the region enclosed by y=e^(5x)+2 y=0 x=0 x=0.7
about the x-axis can be computed using the method of disks or washers via an integral
A=?
b=?
V=?
about the x-axis can be computed using the method of disks or washers via an integral
A=?
b=?
V=?
Answers
Answered by
Steve
using discs, just add up the volumes of tiny slices with radius y and thickness dx:
v = ∫[0,0.7] πy^2 dx
= ∫[0,0.7] π(e^(5x)+2)^2 dx
using shells, you add up the volume of thin cylinders of radius y and height 0.7-x, and thickness dy.
Since the thickness is dy, you have to integrate over y, so you need to express x in terms of y:
x = ln(y-2)/5
Also, for 0<y<2, you have just a solid cylinder of radius 2 and height 0.7
v = π(2^2)(.7) + ∫[2,e^3.5+2] 2πy(0.7-ln(y-2)/5) dy
v = ∫[0,0.7] πy^2 dx
= ∫[0,0.7] π(e^(5x)+2)^2 dx
using shells, you add up the volume of thin cylinders of radius y and height 0.7-x, and thickness dy.
Since the thickness is dy, you have to integrate over y, so you need to express x in terms of y:
x = ln(y-2)/5
Also, for 0<y<2, you have just a solid cylinder of radius 2 and height 0.7
v = π(2^2)(.7) + ∫[2,e^3.5+2] 2πy(0.7-ln(y-2)/5) dy