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.
Best Answer
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.