Problem: Consider a continous curve $y=f(x)$ that passes through $(-1,4), (0,3),(1,4),(2,6),(3,5),(4,6),(5,7),(6,6),(7,4)\text{ and }(9,4).$ Use Simpson's rule to estimate the volume of the solid formed by revolving this curve about the $x-$axis for $[-1.9]$.
My Attempt: As far as I understand, Simpson's rule is used to approximate an integral of the form $$\int_{a}^{b}f(x)dx,$$ however in this problem I do not have an expression for the function. I could split the intervals and obtain a a unique function $y_i$ for it, but I am not sure if that is the right approach. For instance $y_1=f(x)=-x+3$ holds true for all $x\in [-1,0].$ Therefore the volume $$V_1=\pi\int_{-1}^{0}(-x+3)^2dx.$$ And then I could use the Simpson's rule to approximate this integral. Is this the right way to solve this problem?
You don't know the exact form of the function $f(x)$, but you do know enough about it to apply Simpson's rule. Simpson's rule only requires knowing the value at a few locations, and that is exactly what you know.
The gist of Sean Roberson's comment is this, though: You don't want to apply Simpson's rule to $f(x)$ itself. The integral you are trying to estimate is $$\pi\int_0^9 [f(x)]^2~dx$$
I.e., let $g(x) = [f(x)]^2$. Figure out what the values of $g(x)$ are when $x = -1, 0, ..., 9$, and then apply Simpson's rule to $g(x)$.