Riemann Sums

Riemann Sums Exercise 1 : Approximation of Integral Products Let \(f,g\) be continuous on \([a,b]\). For any partition \(P = \{t_0,\ldots,t_n\}\) of \([a,b]\), choose points \(x_i, u_i \in [t_{i-1},t_i]\). Show that sums of the form \[\sum_{i=1}^n f(x_i)g(u_i)(t_i-t_{i-1})\] can be made arbitrarily close to \(\int_a^b fg\) by choosing sufficiently fine partitions \(P\). Exercise 2 : Approximation of Composite Functions Let \(f,g\) be continuous and nonnegative on \([a,b]\). Show that for sufficiently fine partitions \(P\), sums ...