# How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #y=5e^(x)# and #y=5e^(-x)#, x = 1, about the y axis?

##### 2 Answers

Slicing to cylindrical-shell elements for integration gives approximation only. Circular-annular elements are used. To be continued, in the 2nd answer.

#### Explanation:

See graph to see the area that revolves about y-axis ( x = 0 ).

graph{ (y-5(2.718)^x)(y - 5(2.718)^(-x))(x-1+0y)=0[0 1.1 0 13.6]}

The curves meet at A(5, 0).

They meet x = 1 at B( 1, 5 / e ) and C(1, 5e ).

Inversely, the equations are

setting limits for integration with respect to y.

The area ls divided into two parts;

Volume V =

y from 5 / e to 5

Likewise,

with y from 5 to 5 e.

Note that the integrand is the same function of y, for both.So,

with y from 5/e to 5e. Use integration by parts method.

between 5 / e and 5 e

between the limits

between the limits.

I would review my answer for corrections, if any.

The easier cylindrical-shell elements for integration,

applied to right circular cone of height 1 and base radius 1, gives

volume as

.

Continuation, for the 2nd part.

Answer:

#### Explanation:

between the limits y between 5 / e and 5 e. ( Use ln e = 1. )

At the upper limit, the value is

At the lower limit, this becomes

.