Use midpoint rule to estimate the average velocity?

shamieh

Active member
Use the midpoint rule to estimate the average velocity of the car during the first 12 seconds.

i understand the midpoint rule is
$$\displaystyle \frac{b - a}{n}$$

so $$\displaystyle \frac{12}{4} = 3$$
so $$\displaystyle n = 3$$

I also know that

$$\displaystyle \frac{1}{12} \int^{12}_{0} v(t)dt$$

But now I'm stuck... any guidance anyone can offer would be great.

MarkFL

$$\displaystyle \overline{v}(t)=\frac{1}{12}\int_0^{12}v(t)\,dt$$
Using the Midpoint rule to approximate the integral in this expression, with 3 sub-intervals of equal width ($n=3$), we could state:
$$\displaystyle \int_0^{12}v(t)\,dt\approx\frac{12-0}{3}\sum_{k=1}^3\left(v\left(2(2k-1) \right) \right)=4\left(v(2)+v(6)+v(10) \right)$$