Trapeoidal rule
know that Δx=(b-a)/n where the endpoints are x=a and x=b and n=number of subintervals
for this one, b=10 and a=0 and n=10 so (10-0)/10=10/10=1
where f(x) is the function and we start at x=a and end at x=b and starting with the term of x₁ and ending with

, the trapezoidal sum is
![\sum\limits^b_a f(x) dx=\frac{\Delta x}{2}[f(x_1)+2f(x_2)...+2f(x_{n-1})+f(x_n)]](https://tex.z-dn.net/?f=%5Csum%5Climits%5Eb_a%20f%28x%29%20dx%3D%5Cfrac%7B%5CDelta%20x%7D%7B2%7D%5Bf%28x_1%29%2B2f%28x_2%29...%2B2f%28x_%7Bn-1%7D%29%2Bf%28x_n%29%5D)
hmm, just minus the sums and if it is negative, multiply by -1
the integral we aproximating is

that is

[0+2(20)+2(35)+2(48)+2(62)+2(75)+2(85)+2(93)+2(99)+2(106)+111]-

[0+2(18)+2(31)+2(43)+2(58)+2(68)+2(79)+2(86)+2(93)+2(95)+96]=59.5
B. since it was positive, sappho's car traveled 59mi farther than homer's car
C. average velocity is all the velocities added together
wait, midpoints?
hmm, ok, so we want 5 intervals
so we take average of each section then average them
5 sections
ah, got it
for midpoint, we don't generate our own data
example, from t=0 to t=2, the midpoint value is 20 because it is at t=1
we do the intervals
0 to 2, 2 to 4, 4 to 6, 6 to 8, 8 to 10
midpoints are 20, 48, 75, 93, 106
so we do
(1/5)(20+48+75+93+106)=68.4
the average velocity is 68.4 mph