§5.6 Numerical integration

Introduction

The Trapezoid Rule

Comparing the Midpoint and Trapezoid Rules

Simpson's Rule

As we begin to investigate ways to approximate definite integrals, it will be insightful to compare results to integrals whose exact values we know. To that end, the following sequence of questions centers on $\int_0^3 x^2 \, dx$.

Use https://gvsu.edu/s/a9 with the function $f(x) = x^2$ on the window of $x$ values from $0$ to $3$ to compute $L_3$, the left Riemann sum with three subintervals.

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

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 $M_8$.

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

. Similarly, we compute the error of the midpoint rule approximation with 8 subintervals by the formula

. Based on your work in (a) and (b) above, compute $E_{T,4}$, $E_{T,8}$, $E_{M,4}$, $E_{M,8}$.

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

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

Plot the given data on the set of axes provided in the figure, with time on the horizontal axis and the velocity on the vertical axis.

What definite integral will give you the exact distance the car traveled on $[0,1.8]$?

Estimate the total distance traveled on $[0,1.8]$ by computing $L_3$, $R_3$, and $T_3$. Which of these under-estimates the true distance traveled?

Estimate the total distance traveled on $[0,1.8]$ by computing $M_3$. Is this an over- or under-estimate? Why?

Using your results from (c) and (d), improve your estimate further by using Simpson's Rule.

What is your best estimate of the average velocity of the car on $[0,1.8]$? Why? What are the units on this quantity?