Question

let r(x) be the radius of a round pipe that drains water from a dam, where x is measured in feet from the dam. Which choice best explains the meaning of pi integral 10000-30000 (r(x))^2 dx

Answer 1

\[\pi\int\left(10,000-30,000\left[r(x)\right]^2\right)dx?\]
Judging by what little information you've provided, r(x) is apparently a function of the radius of the pipe with respect to its distance from the dam. It's not apparent whether r(x) is increasing, decreasing, or staying constant.

I get the feeling that the integral I've written isn't what you want to say. The reason I say that is that if you had the exponent out side the set of parentheses, it would look like the integral represents the volume of water contained by the pipe over a certain distance.

So for some interval, say [a, b], the integral would represent the volume of water contained in the pipe over a distance of (b - a).