# The Fundamental Theorem of Calculus

### Topics

## Introduction to The Fundamental Theorem Of Calculus - At A Glance:

Let *v*(*t*) be a velocity function on the time interval [*a*,*b*]. The function *v*(*t*) describes the movement of something–maybe a car, maybe an emu, maybe a banana slug. The banana slug is at some starting position when *t* = *a*, travels some distance from *t* = *a* to *t* = *b*, and is at some ending position when *t* = *b*.

If *v*(*t*) ≥ 0 on [*a*,*b*], then is positive and is the distance travelled between time *t* = *a* and time *t* = *b*.

We could also say

If *v*(*t*) ≤ 0 on [*a*,*b*], then is negative and is the distance travelled between time *t* = *a* and *t* = *b*, but in the "opposite" direction.

In other words,

If *v*(*t*) is sometimes negative and sometimes positive on [*a*,*b*], then is the distance travelled in the positive direction

minus the distance travelled in the negative direction

We can also describe this by

Regardless of whether the velocity function is positive, negative, or a little bit of each,

We're going to introduce a new function *s*(*t*) so that we can write this equation more compactly. Let

*s*(*t*) = position at time *t*.

So *s*(*a*) = position at time *t* = *a*, *s*(*b*) = position at time *t* = *b*, etc.

On the interval [*a*,*b*], *s*(*a*) is the position of the car or cheetah at the start of the interval and *s*(*b*) is the position of the car or cheetah at the end of the interval. The change in position of the car or cheetah over the interval [*a*,*b*] is

(ending position) – (starting position) = *s*(*b*) – *s*(*a*)

However, the change in position of the car or cheetah over the interval [*a*,*b*] is also . This means

We'd like to point out that velocity is the derivative of position (change in position with respect to time). In symbols,

*v*(*t*) = *s*'(*t*).

For the following examples and exercises, assume *s*(*t*) is the position function and *v*(*t*) = *s*'(*t*) is the velocity function.

The equation

describes a relationship between the three values , *s*(*a*), and *s*(*b*). Most problems involving this equation will give you two of these values and ask you to find the third.

You could be given the value of the integral and one of the values *s*(*a*) or *s*(*b*).

We've been working with the equation

in this form because this is how the Fundamental Theorem of Calculus is usually given. However, we can also rearrange the equation

by adding *s*(*a*) to both sides to get this:

This rearranged equation says that if you take the starting position (*s*(*a*)) and add the change in position you get the ending position

(*s*(*b*)).

Some problems are easier if we use the rearranged equation, especially those where we're given *s*(*a*) and and asked to find *s*(*b*).

#### Example 1

A particle moving along the where is the particle when |

#### Example 2

A cat climbing up and down a tree ends at height 3 feet after 10 seconds. If feet, how high in the tree was the cat
at time |

#### Example 3

Let (a) If , that means we had to take 3 away from (b) If , that means we had to add 7 to |

#### Example 4

A bicyclist is 10 miles from home after 3 hours of riding and 30 miles from home after 5 hours of riding. If her velocity is given by the function |

#### Example 5

If |

#### Exercise 1

Let *s*(*t*) be a position function with velocity function *v*(*t*) = *s*'(*t*).

(a) If *s*(9) = 4 and , what is *s*(12)?

(b) If *s*(3) = -2 and , what is *s*(5)?

#### Exercise 2

A koala is climbing up and down its tree with velocity *v*(*t*) feet per minute, where positive values of *v*(*t*) indicate the koala is climbing up the tree.

At t = 0 minutes the koala is 5 feet above ground.

(a) If feet, how high is the koala after *t* = 4 minutes?

(b) If feet, how high is the koala at *t* = 7 minutes (use part (a))?

#### Exercise 3

A hummingbird flies away from its feeder with velocity *v*(*t*) feet per second and position *s*(*t*) feet away from its feeder.

(a) If *s*(4) = 5 feet and feet, find the hummingbird's distance from the feeder at time *t* = 0.

(b) If *s*(10) = 16 feet and feet, find *s*(5).

#### Exercise 4

Let *s*(*t*) be a position function with velocity function *v*(*t*) = *s*'(*t*).

(a) If and *s*(9) = -3, what is *s*(8)?

(b) If and *s*(7) = -4, what is *s*(3)?

(c) Given (a) and (b), what is ?

(d) Given (a) and (b), what is ?

#### Exercise 5

Let *s*(*t*) be a position function with velocity function *v*(*t*) = *s*'(*t*).

(a) If *s*(2) = 7 and *s*(9) = 13, what is ?

(b) If *s*(2) = 13 and *s*(9) = 7, what is ?

(c) If *s*(2) = 13 and *s*(9) = 7 then what is ? (hint: one of the properties of integrals says how to change the limits of integration)