Finding the volume of the body by rotating two intervals around the x-axis

Question

Find the volume of the body that is created by the rotation of these intervals around the x-axis

$$0\leq x \leq {\pi\over 2}$$ $$0\leq y \leq (e^t\times sin(t))$$

I have no idea on where to begin or even know how to do this, I don't want an answer directly but please give me a direction on what to do to solve this problem, I don't really understand the question.

Am I supposed to create a function of these two and then add them together using the disc or shell method?

I've been stuck on this problem for a few days and can't find any resources for help.

Answer 1

Here is a sketch of the graphs:

Let's see what happens if we rotate it about the $x$-axis:

Will discs or shells make the following solid?

They are discs, right? So we are adding together discs with radii $r(x)=\sin(x)e^x$. This makes the area of each disc $\pi\left(\sin(x)e^x\right)^2$. Now adding all of these disc areas gives you $$\int_0^{\pi/2}\pi\left(\sin(x)e^x\right)^2\,dx.$$ Can you take it from here?