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

Answer 1

since e^(lnπ) = π,
sin(π) is the first time the curve intersects the x-axis.

So, the volume, using discs of thickness dx is

v = ∫[0,lnπ] πr^2 dx
where r=y=sin(e^x)
v = π∫[0,lnπ] sin^2(e^x) dx

Now, sin(e^x) is not integrable using elementary functions. I guess you'll have to approximate it using some Riemann sum.