Activity 5.6.2

In this activity, we explore the relationships among the errors generated by left, right, midpoint, and trapezoid approximations to the definite integral $\int_1^2 \frac{1}{x^2} \, dx$.

1. Use the First FTC to evaluate $\int_1^2 \frac{1}{x^2} \, dx$ exactly.

2. Use appropriate computing technology to compute the following approximations for $\int_1^2 \frac{1}{x^2} \, dx$: $T_4$, $M_4$, $T_8$, and $M_8$.

3. Let the error that results from an approximation be the approximation's value minus the exact value of the definite integral. For instance, if we let $E_{T,4}$ represent the error that results from using the trapezoid rule with 4 subintervals to estimate the integral, we have $$\begin{gathered} E_{T,4} = T_4 - \int_1^2 \frac{1}{x^2} \, dx \end{gathered}$$. Similarly, we compute the error of the midpoint rule approximation with 8 subintervals by the formula $$\begin{gathered} E_{M,8} = M_8 - \int_1^2 \frac{1}{x^2} \, dx \end{gathered}$$. Based on your work in (a) and (b) above, compute $E_{T,4}$, $E_{T,8}$, $E_{M,4}$, $E_{M,8}$.

4. Which rule consistently over-estimates the exact value of the definite integral? Which rule consistently under-estimates the definite integral?

5. What behavior(s) of the function $f(x) = \frac{1}{x^2}$ lead to your observations in (d)?