Question

Y = 7ex, y = 7e−x, x = 1; about the y-axis

Answer 1

A "typical" cylindrical shell of wall thickness dx at x=x will have the length (in the y-direction) L = (e^x – e^-x). 

Rotating this shell about the y-axis generates a volume dV = 2.π.x.L = 2.π.x.(e^x – e^-x).dx 

Total volume V of all such cylindrical shells in the given range = 2.π.∫x.(e^x – e^-x).dx 

= 4.π.[x.cosh(x) – sinh(x)] from 0 to 1. Now insert the limits and we obtain: 

= 4.π.[1.543 – 1.175] = 4.624 cubic units