Volume of a solid? $$
\begin{array}{rrrrrrrr}
x & 0 & 0.5 & 1.0 & 1.5 & 2.0 & 2.5 & 3.0
\\
f(x) & 2 & 1.3 & 0.9 & 0.6 & 0.7  & 1.1 & 1.9
\end{array}
$$
Find a formula for the volume $V$ of the solid whose base is the region bounded by $y = f(x)$, the $x$-axis, and the line $x = 3$ and its cross-sections perpendicular to the $x$-axis are semicircles.**
So, I plotted the points and got a graph that looks something like this: 

Now to start on actually solving the problem.
So I figure that we should break the region up into a small $dx$ pieces, and just sum up all of these pieces using an integral.
However, I'm having trouble figuring our what the area of each piece will be. Any help?
 A: If I am understanding the geometry right, the semicircles have radius $f(x)/2$, and therefore area $(\pi/8)(f(x))^2$. Call this area $g(x)$.
We want $\int_0^3 g(x)\,dx$. But since we do not have a formula for $f(x)$, we do not have one for $g(x)$.  
You are now very likely intended to use a numerical integration method.
From the given values of $f(x)$, you can compute $g(x)$ at $0$, $0.5$, and so on up to $3$.  So you have a subdivision of the interval of integration into $6$ equal parts of width $\Delta x$, where $\Delta x=0.5$.  Now it is straightforward to use the Trapezoidal Rule, or Simpson's Rule. If numerical integration has not been discussed yet, you can use a Riemann sum to approximate the integral. The error is then likely to be larger than if you use Simpson's Rule, or the Trapezoidal Rule.   
Remark: To cut down on the wear and tear on your calculator, and on the likelihood of keying errors, you should alter the computational procedure slightly. Let $h(x)=(f(x))^2$. Then we want
$\frac{\pi}{8}\int_0^3 h(x)\,dx$.  Now we only need to find the $(f(x_i))^2$, process using say Simpson's rule, and multiply by $\pi/8$ to get the final answer.  Squaring of $2$, $1.3$, and so on up to $1.9$ can even be done in one's head.
