Evaluate the following definite integrals by finding the limits of the upper or lower sums.

• [math]\int_0^1 x^2 \; dx[/math]
• [math]\int_0^2 2x \; dx[/math]
• [math]\int_1^3 (x+1) \; dx[/math]
• [math]\int_0^1 (3x^2 + 1) \; dx[/math].

For each of the integrals in Problem Exercise, draw the region whose area is given by the integral.

Let [math]f[/math] be the step function defined by [math]f(x) = i[/math], if [math]i-1 \lt x \leq i[/math], for every integer [math]i[/math]. Draw the graph of [math]f[/math] and compute the following integrals. (Hint: These problems are neither hard nor long. They require an understanding of the definition of integrability and possibly some ingenuity.

• [math]\int_1^2 f[/math]
• [math]\int_0^3 f[/math]
• [math]\int_{-1}^3 f[/math]
• [math]\int_{-2}^7 f[/math].

Every constant function is both increasing and decreasing. A stronger condition, which excludes constant functions, is obtained by defining [math]f[/math] to be strictly increasing if

for every [math]x[/math] and [math]y[/math] in the domain of [math]f[/math]. The companion definitions of what it means for a function to be strictly decreasing, strictly increasing on an interval, etc., should be obvious. Using the Mean Value Theorem, prove that if a differentiable function [math]f[/math] satisfies the inequality [math]f^\prime(x) \gt 0[/math] for every [math]x[/math] in an interval [math]I[/math], then [math]f[/math] is strictly increasing on [math]I[/math].

Prove the converse of Theorem \ref{thm 4.3.2}; i.e., if [math]f[/math] is integrable over [math][a,b][/math], then [math]\lim_{n\goesto\infty} (U_n - L_n) = 0[/math]. (This is a difficult problem.)