The region in the first quadrant bounded by the x-axis, the line x = ln(π), and the curve y = sin(ex) is rotated about the x-axis. What is the volume of the generated solid?

Answer 1

integral from x = 0 to x = ln (pi)
of
pi y^2 dx = pi sin^2 e x dx ?

= pi [ x/2 - sin (2 e x)/4e ] at x = ln pi - at x = 0
but 0 at x = 0 so
= pi [ (ln pi) /2 - sin (2 e ln pi)/4e ]
thing is ln pi is just about 1.145 and e is about 2.7183
so 2 e ln pi is about 6.22
which is 2 pi :)
and we know sin 2 pi = 0
so we really have
pi (1.145)/ 2 which is about 1.8