(2) Recall that, on an interval [a, b], we define the definite integral to be
$\int_a^b f(x)dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x$.
Express the following limit as a definite integral:
$\lim_{n \to \infty} \sum_{i=1}^n cos^3(14\pi x_i) \Delta x$ on the interval [0, 1].
Solution.
(3) Suppose that $f(x)$ and $g(x)$ are functions which satisfy
$\int_0^1 f(x)dx = 2$, $\int_1^2 f(x)dx = 1$, and $\int_0^2 g(x) = -1$.
Use properties of definite integrals to compute the following definite integral:
$\int_0^2 3f(x) - 2g(x)dx$.
Be sure to write all relevant steps, and not just the answer.
Solution.
