§5.6 Numerical integration

Introduction

The Trapezoid Rule

Comparing the Midpoint and Trapezoid Rules

Simpson's Rule

For $f(x) = \frac{1}{x^2}$ on $[1,2]$, which is concave up, the trapezoid and midpoint rules satisfy

For $\int_0^3x^2\,dx$ with three equal subintervals, the left endpoint sum $L_3$ equals

Simpson's rule combines the midpoint and trapezoid estimates on matching subintervals as

With exact value $\frac{1}{2}$ and computed values $T_4 = 0.50899$ and $M_4 = 0.49555$, the errors $E_{T,4}$ and $E_{M,4}$ are

A braking car's velocity is positive throughout $[0,1.8]$. The exact distance traveled is given by

With $S_6 = 140.8$ feet as the best distance estimate on $[0,1.8]$, the best average velocity estimate is

A braking car's velocity table with three panels gives $L_3 = 165.6$ feet and $R_3 = 105.6$ feet. Compute the trapezoid estimate $T_3$ in feet.

With $M_3 = 143.4$ feet and $T_3 = 135.6$ feet for the braking car, compute Simpson's estimate $S_6$ in feet.

By the First FTC, evaluate $\int_1^2\frac{1}{x^2}\,dx$ as a decimal.

Using the Simpson distance estimate $140.8$ feet over $1.8$ seconds, compute the average velocity, rounded to two decimals.

For the braking car whose velocity graph is concave down, the midpoint estimate $M_3 = 143.4$ feet is an overestimate of the true distance.