what is the interval for this cylindrical shell problem?

Question

For the question attached, my textbook integrates from the interval 0 to 1, but shouldn't it be integrated from the interval of -1 to 1?

Answer 1

No. Remember that the shell method takes the volume from lots of cylindrical husks, and sums them up to produce the volume of the entire shape. To understand why we don't integrate over the bounds of the final solid, we need to take a look at the mechanics of how it works.

Note that the volume of a representative shell will be given by $$\begin{align}\Delta V &= 2\pi(\text{average radius})(\text{height})(\text{thickness}) \\ &=2\pi r(x)h(x)\Delta x.\end{align}$$

The average radius is from the axis of rotation in question, and remember that radius of a circle is half of the diameter. This is the key point. The $$r(x)h(x)\Delta x$$ produces the area for a thin rectangle away from the axis of rotation with height $h(x)$. So what does the $2\pi$ do? The $2\pi$ spins this around in a circle about the axis of rotation (remember the formula for circumference of a circle?) which is how you'll hit the $[-1,0]$ portion of the shape (again, because radius is half of the diameter).