Chapter 5, Problem 1SYU

To determine

To indicate:

Whether the statement “If the velocity is positive, then the total distance traveled is the area under the velocity curve” is true or false.

Answer to Problem 1SYU

The statement is true.

Explanation of Solution

Given information:

The statement “If the velocity is positive, then the total distance traveled is the area under the velocity curve”.

Concept used:

$\text{Distance}\text{traveled}=\text{Velocity}\times \text{Time}$

Area under the curve $V\left(t\right)$ on any interval $\left[a,b\right]$ is given by

${\displaystyle \underset{a}{\overset{b}{\int }}V\left(t\right)}dt$

Here, $V\left(t\right)$ is the velocity function and $dt$ is time interval.

If we plot a graph of velocity against time, we get an area between the velocity curve and the horizontal axis.

From the above graph we see that the total distance traveled is the area under the velocity curve.