Use the Midpoint Rule with [math]n=4[/math] to compute approximations to the following integrals. In \ref{ex8.3.1a}, \ref{ex8.3.1b}, \ref{ex8.3.1c}, \ref{ex8.3.1d}, and \ref{ex8.3.1e} compare the result obtained with the true value.

• lab{8.3.1a} [math]\int_0^1 (x^2+1)\;dx[/math]
• lab{8.3.1b} [math]\int_{-1}^3 (6x-5)\;dx[/math]
• lab{8.3.1c} [math]\int_1^3 \frac1{x^2} dx[/math]
• lab{8.3.1d} [math]\int_0^3 \frac1{1+x} dx[/math]
• lab{8.3.1e} [math]\int_0^3 \sqrt{1+x}\;dx[/math]
• [math]\int_0^{2\pi} \sin^2x\;dx[/math]
• [math]\int_0^1 e^{-x^2} dx[/math]
• lab{8.3.1h} [math]\int_0^1 \sqrt{1+x^3}\; dx[/math].

\ref{ex8.3.1a} through \ref{ex8.3.1h}. Compare [math]M_4[/math], the Midpoint Approximation computed in Problem Exercise, to [math]T_4[/math], the corresponding Trapezoid Approximation.

\ref{ex8.3.1a} through \ref{ex8.3.1h}. Use Simpson's Rule with [math]n=4[/math] to compute an approximation to the corresponding definite integral in Problem Exercise.

Show geometrically that, if the graph of [math]f[/math] is concave up at every point of the interval [math][a,b][/math], then the Midpoint Approximation is too small and the Trapezoid Approximation is too big; i.e.,

Do Problem Exercise analytically by using the remainder formulas \ref{thm 8.2.4} and \ref{thm 8.3.3}.

Show that Simpson's Approximation is the weighted average of the Trapezoid Approximation and the Midpoint Approximation. Specifically, for any even positive integer [math]n=2m[/math], show that

Prove Theorem \ref{thm 8.3.5}, i.e., if [math]f[/math] is integrable over [math][a,b][/math], then

by showing that it is a direct corollary of the result of Problem Exercise and the two corresponding theorems, [math]\lim_{n\goesto\infty}T_n=\int_a^b f[/math] and [math]\lim_{n\goesto\infty}M_n=\int_a^b f[/math].

For each of the following integrals and each of the three methods of numerical integration (Trapezoid Rule, Midpoint Rule, and Simpson's Rule), find the smallest integer [math]n[/math] such that the error obtained is less that [math]10^{-4}[/math]. As the basis for finding [math]n[/math], use Theorems \ref{thm 8.2.4}, \ref{thm 8.3.3}, and \ref{thm 8.3.6}.

• [math]\int_1^4 \left(\frac1{2x^2} + \frac{x^2}2\right)\;dx[/math]
• [math]\int_0^2 \frac1{2x+1} dx[/math].

This problem is analogous to Problem Exercise. Show that, for any positive integer [math]n[/math],

Suppose that the graph of [math]f[/math] is concave up at every point of the interval [math][a,b][/math].

• Using the results of Problems Exercise and Exercise, show that
  [[math]] T_{2n} - (T_n - T_{2n}) \lt \int_a^b f \lt T_{2n} , [[/math]]
  for every positive integer [math]n[/math].
• lab{8.3.10b} Hence show that the error [math]|\int_a^b f-T_{2n}|[/math] in the Trapezoid Approximation satisfies
  [[math]] \left| \int_a^b f-T_{2n}\right| \lt |T_n - T_{2n}| . [[/math]]
• Show that \ref{ex8.3.10b} also holds if the graph of [math]f[/math] is concave down at every point of [math][a,b][/math].