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Use midpoint rule to estimate the average velocity?

shamieh

Active member
Sep 13, 2013
539
Use the midpoint rule to estimate the average velocity of the car during the first 12 seconds.

Click here to see the graph from my book.

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

Administrator
Staff member
Feb 24, 2012
13,775
The average velocity would be given by (as you stated):

\(\displaystyle \overline{v}(t)=\frac{1}{12}\int_0^{12}v(t)\,dt\)

Using the Midpoint rule to approximate the integral in this expression, with 3 sub-intervals of equal width ($n=3$), we could state:

\(\displaystyle \int_0^{12}v(t)\,dt\approx\frac{12-0}{3}\sum_{k=1}^3\left(v\left(2(2k-1) \right) \right)=4\left(v(2)+v(6)+v(10) \right)\)

Do you see that we evaluate the velocity function at the midpoint of each sub-interval?

Now you just need to read the needed values from the given graph.