# Speed of a runner

I am trying to do this homework problem for calculus. It is an intro to integrals and I have no idea what I am doing wrong.

The speed of a runner is increase steadily during the first $3$ seconds of a race. Her speed at half seconds intervals is given in the table. Find lower and upper estimates for the distance she traveled during these $3$ seconds.

$$\begin{matrix} t(\mathrm{s}) & & 0 & .5 & 1 & 1.5 & 2 & 2.5 & 3 \\ v(\mathrm{ft}/\mathrm{s}) & & 0 & 6.2 & 10.8 & 14.9 & 18.1 & 19.4 & 20.2 \end{matrix}$$

I drew two graphs of this and added the rectangles for the estimate and I got two very different looking estimates, one obviously over and one obviously lower. If I understand this right the over estimate is calculated like so $$.5(6.2) + .5(10.8) + \cdots ,$$ which seems correct to me.

The under estimate is calculated $0 \cdot 0 + \text{the rest of the series}$.

I don't see what makes these two different but neither are correct numbers and I get the same for both which I know it isn't right.

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Your under-estimate should be multiplying by the length of the intervals in the same way your over-estimate did. So you will have $$0.5(0)+0.5(6.2)+\cdots+0.5(19.4)$$ which will not be the same as your over-estimate, which is $$0.5(6.2)+0.5(10.8)+\cdots+0.5(20.2).$$

The difference is that you are using the speed at the start of the time interval rather than the (faster) speed at the end.

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I don't undestand the difference. I am going to end up with the same numbers, just on the left interval one I start with a times 0. – user138246 Oct 31 '11 at 23:39
On the left interval sum, you won't include the very last term. You should have the same number of terms (6) in each estimate. – Michael Joyce Oct 31 '11 at 23:42
You should have six intervals for each. Based on the values you are given, you assume that the speed at the left end is the minimum speed for that interval, and the right is the maximum. So when you want to underestimate, you multiply the length of each interval (0.5), by its minimum speed (the left). To overestimate you multiply the interval length (0.5) by the maximum speed (the right). You always have six intervals, but some terms will be different in each sum. – AMPerrine Oct 31 '11 at 23:49
You never multiply by $0, 0.5, 1,$ etc. but by $(0.5-0),(1-0.5),(1.5-1),$ etc.--the lengths of the intervals, ie. $\Delta x$. If you check your book again, you should see that right uses every point except the first and left uses every point except the last. If it does not, then that is a mistake in the book. Right must use only the points on the right end of the intervals, and left only the points on the left. Your graph sounds correct. – AMPerrine Nov 1 '11 at 0:07
I just can't get this to work, I can never get the same numbers. The book says I should be able to get the same number. – user138246 Nov 1 '11 at 0:15