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 [tex]x_n[/tex], the trapezoidal sum is
[tex]\sum\limits^b_a f(x) dx=\frac{\Delta x}{2}[f(x_1)+2f(x_2)...+2f(x_{n-1})+f(x_n)][/tex]
hmm, just minus the sums and if it is negative, multiply by -1
the integral we aproximating is [tex] \int\limits^{10}_0 {vS(t)-vH(t)} \, dt [/tex]
that is
[tex]\frac{1}{2}[/tex][0+2(20)+2(35)+2(48)+2(62)+2(75)+2(85)+2(93)+2(99)+2(106)+111]-[tex]\frac{1}{2}[/tex][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
