The volume of the solid obtained by rotating the region bounded by y=e^x, y=lnx, x=1, and x=5 about the line y-axis can be computed using the method of cylindrical shells

Answer 1

Answer by Yefim F. shows the antiderivative of xe^x and xlnx which are determined using integration by parts.

Here is a graph confirming the solution:

desmos.com/calculator/4maeifayer

Answer 2

v = 2π∫₁⁵x(e^x - lnx)dx = 2π(xe^x - e^x - x²/2lnx + x²/4)₁⁵ = 3603.625