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

Question

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

Answer 1

You know that e^ln(n) = n

so y(ln(n)) = sin(n)

That means that the solid of revolution has volume

v = ∫[0,ln(n))] π sin²(e^x) dx

Now, that cannot be evaluated using elementary functions, but you can use your favorite numerical method to find that it comes out to

π/2 (ln(n)+Ci(2)-Ci(2n))

where Ci(x) is the cosine integral function.

If we take n=π so we get the area under one full arch of the curve, the volume is approximately 2.5