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 \(\int_1^2 \frac{1}{x^2} \, dx\text{.}\)
(a)
Use the First FTC to evaluate \(\int_1^2 \frac{1}{x^2} \, dx\) exactly.
(b)
Use appropriate computing technology to compute the following approximations for \(\int_1^2 \frac{1}{x^2} \, dx\text{:}\) \(T_4\text{,}\) \(M_4\text{,}\) \(T_8\text{,}\) and \(M_8\text{.}\)
(c)
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{equation*}
E_{T,4} = T_4 - \int_1^2 \frac{1}{x^2} \, dx \text{.}
\end{equation*}
Similarly, we compute the error of the midpoint rule approximation with 8 subintervals by the formula
\begin{equation*}
E_{M,8} = M_8 - \int_1^2 \frac{1}{x^2} \, dx\text{.}
\end{equation*}
Based on your work in (a) and (b) above, compute \(E_{T,4}\text{,}\) \(E_{T,8}\text{,}\) \(E_{M,4}\text{,}\) \(E_{M,8}\text{.}\)
(d)
Which rule consistently over-estimates the exact value of the definite integral? Which rule consistently under-estimates the definite integral?
(e)
What behavior(s) of the function \(f(x) = \frac{1}{x^2}\) lead to your observations in (d)?
