Integrating the Velocity Function Examples

Suppose JoJo was driving 30 mph at 5 pm and 40 mph at 7 pm. How far did she travel between 5 and 7 pm?

If JoJo was driving 30 mph at the beginning and 40 mph at the end of the time interval, we can pretend she's been going 35 mph the whole time. Then we estimate her distance to be

35(2) = 70 miles.

The velocity of a racing snail, in feet per second, is given below.

How far does the racing snail travel

(a) from t = 0 to t = 10 seconds?

(b) from t = 10 to t = 15 seconds?

(c) from t = 15 to t = 25 seconds?

(d) from t = 0 to t = 25 seconds?

(a) The distance the snail travels from t = 0 to t = 10 seconds is given by

We can calculate this integral by looking at the graph:

The area between v(t) and the t axis on [0,10] is 200, so the snail travels 200 feet between t = 0 and t = 10 seconds.

(b) The distance the snail travels from t = 10 to t = 15 seconds is

To calculate this integral we look at the graph:

The distance the snail travels from t = 10 to t = 15 seconds is

(c) From t = 15 to t = 25 seconds the snail travels

as we can see from the graph:

(d) The area between v(t) and the t-axis on [0,25] adds up to 525, as we can see from the graph:

This means from t = 0 to t = 25 the snail travels

For a sanity check, if we add up the distance travelled on each time interval we also get 525 feet:

A bug moves along a numberline with velocity v(t) in units per second given by the graph below.

If the bug is at 3 on the numberline when t = 0 seconds, where is the bug when

(a) t = 3 seconds?

(b) t = 5 seconds?

First, look at the areas between the graph of v(t) and the t-axis:

(a) From t = 0 to t = 3 the bug travelled

Since the bug started at 3, after traveling 12 units it ends up at 15.

(b) We know the bug is at 15 when t = 3 seconds. Looking at the graph of v(t) we can see that from t = 3 to t = 5 seconds the bug travels

This means when t = 5, the bug is at 19.