Chapter 7.CR, Problem 56CR

In Exercises 32 and 33 of Section $7.3$ on Area and the Definite Integral, you calculated the distance that a car traveled by estimating the integral ${\displaystyle {\int }_0^{T}v\left(t\right)dt}$.

a. Let $s\left(t\right)$ represent the mileage reading on the odometer. Express the distance traveled between $t=0$ and $t=T$ using the function $s\left(t\right)$.

b. Since your answer to part a and the original integral both represent the distance traveled by the car, the two can be set equal. Explain why the resulting equation is a statement of the Fundamental Theorem of Calculus.