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?

First, it would be good to know that the area bounded by the curve and the x-axis is convergent to begin with.



Let , so that , and the integral is equivalent to



The integrand is continuous everywhere except , but that's okay because we have . This means the integral is convergent - great! (Moreover, there's a special function designed to handle this sort of integral, aptly named the "sine integral function".)

Now, to compute the volume. Via the disk method, we have a volume given by the integral



By the same substitution as before, we can write this as



The half-angle identity for sine allows us to rewrite as



and replacing , , we have



Like the previous, this require a special function in order to express it in a closed form. You would find that its value is



where is the Euler-Mascheroni constant and denotes the cosine integral function.

put like terms together.orremember that the 45 becomes negative when moved to the other side.orthe minuses cancel out.