The velocity of a pink elephant, in feet per minute, is given by the function v(t) below.
How far does the elephant travel
(a) from t = 0 to t = 10?
(b) from t = 10 to t = 25?
(c) from t = 25 to t = 30?
(d) from t = 0 to t = 30?
We're going to need to know the area between the graph of v(t) and the t-axis on each time interval, so let's just figure those areas out now:
(a) From t = 0 to t = 10 the pink elephant travels
(b) From t = 10 to t = 25 the pink elephant travels
(c) From t = 25 to t = 30 the pink elephant travels
(d) There are two ways to figure out how far the elephant travels from t = 0 to t = 30. One way is to look at the graph again, add up all the areas, and get
The other way is to remember the property of integrals that says
The integrals on the right-hand side were calculated in (a)-(c), and now we just add them all up: