Draw the graph of the function [math]f[/math] defined by [math]f(x) = \frac1x[/math], and answer the following questions.

• </math>?
• Is [math]f[/math] bounded on the open interval [math](2,5)[/math]?
• Does [math]f[/math] have an upper bound on the interval [math](0,2)[/math]? If so, give one.
• Does [math]f[/math] have a lower bound on the interval [math](0,2)[/math]? If so, give one.

If the number [math]M[/math] is the least upper bound of the set of all numbers [math]f(x)[/math] for [math]x[/math] lying in an interval [math]I[/math], we say simply that [math]M[/math] is the least upper bound of [math]f[/math] on [math]I[/math]. A similar remark holds for the greatest lower bound. Draw the graph of the function [math]f[/math] defined by [math]f(x) = \frac1{x-1}[/math], and answer the following questions.

• What is the least upper bound of [math]f[/math] on the closed interval [math][2,3][/math]?
• What is the greatest lower bound of [math]f[/math] on [math][2,3][/math]?
• What are the least upper bound and greatest lower bound of [math]f[/math] on the open interval [math](2,3)[/math]?
• What is the greatest lower bound of [math]f[/math] on the interval [math](1,2)[/math]?

Compute the upper and lower sums [math]U_\sigma[/math] and [math]L_\sigma[/math] in each of the following examples.

• [math]f(x) = \frac1x[/math], [math][a,b] = [1,4][/math], and [math]\sigma = \{1,2,3,4\}[/math].
• [math]f(x) = \frac x2[/math], [math][a,b] = [0,2][/math], and [math]\sigma = \{0, \frac13, \frac23, 1, \frac43, \frac53, 2\}[/math].
• [math]g(x) = x^2 + 1[/math], [math][a,b] = [0,1][/math], and [math]\sigma = \{x_0,x_1,x_2,x_3,x_4,x_5\}[/math], where [math]x_i = \frac i5, i = 0, \ldots, 5[/math].
• [math]g(x) = x^3[/math], [math][a,b] = [-1,1][/math], and [math]\sigma = \{-1, -\frac12, 0, \frac12, 1\}[/math].

True or false, and give your reason: If a function [math]f[/math] is continuous at every [math]x[/math] is a closed interval [math][a,b][/math], then [math]f[/math] has both a least upper bound and a greatest lower bound on [math][a,b][/math].

Assume that the function [math]f[/math] defined by [math]f(x) = x^2 + 1[/math] is integrable over the interval [math][0,1][/math]. Using the partition [math]\sigma = \{0, \frac15, \frac25, \frac35, \frac45, 1\}[/math], show that

Assuming that the function [math]g[/math] defined by [math]g(x) = 2x[/math] is integrable over the interval [math][0,2][/math], use the partition [math]\sigma = \{0, \frac12, 1, \frac32, 2\}[/math] to show that

Assume that the function [math]x^2[/math] is integrable over the interval [math][0,1][/math]. Using the partition [math]\sigma = \{x_0, \ldots, x_n\}[/math], where [math]n=10[/math] and [math]x_i = \frac{i}{10}[/math], for [math]0,\ldots,10[/math], prove that

Compute the definite integral [math]\int_a^b f = \int_a^b f(x) \; dx = \int_a^b f(t) \; dt[/math] in each of the following examples. Assume that [math]f[/math] is integrable, and use Theorem \ref{thm 4.1.4} and the standard formulas for the areas of simple plane figures. In each case, draw the graph of [math]f[/math] and shade the region [math]P[/math].

• [math]\int_{-1}^1 f[/math], where [math]f(x) = \sqrt{1-x^2}[/math].
• [math]\int_1^2 f(t) \; dt[/math], where [math]f(t) = t-1[/math].
• [math]\int_0^2 2x \; dx[/math]
• [math]\int_0^1 (5-2x) \; dx[/math]
• [math]\int_{-1}^1 |x| \; dx[/math].

It is stated in this section that the first condition for integrability is always satisfied: If [math]f[/math] is bounded on [math][a,b][/math], then there exists a real number [math]J[/math] such that [math]L_\sigma \leq J \leq U_\tau[/math] for any two partitions [math]\sigma[/math] and [math]\tau[/math] of [math][a,b][/math].

• Show that one such number is the least upper bound of all the lower sums [math]L_\sigma[/math]. (This number is called the lower integral of [math]f[/math] from [math]a[/math] to [math]b[/math].)
• Show that another possibility is the greatest lower bound of all the upper sums [math]U_\tau[/math]. (This number is the upper integral of [math]f[/math] from [math]a[/math] to [math]b[/math].)
• Show that [math]f[/math] is integrable over [math][a,b][/math] if and only if the lower integral from [math]a[/math] to [math]b[/math] equals the upper integral, and that if the lower integral equals the upper then their common value is [math]\int_a^b f[/math].