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

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