5.4 Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, WashingtonPhoto by...

5.4 Fundamental Theorem of Calculus

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1998

Morro Rock, California

If you were being sent to a desert island and could take only one equation with you,

x

a

df t dt f x

dx

might well be your choice.

Here is my favorite calculus textbook quote of all time, from CALCULUS by Ross L. Finney and George B. Thomas, Jr., ©1990.

The Fundamental Theorem of Calculus, Part 1

If f is continuous on , then the function ,a b

x

aF x f t dt

has a derivative at every point in , and ,a b

x

a

dF df t dt f x

dx dx

x

a

df t dt f x

dx

First Fundamental Theorem:

1. Derivative of an integral.

a

xdf t dt

xf x

d

2. Derivative matches upper limit of integration.

First Fundamental Theorem:

1. Derivative of an integral.

a

xdf t dt f x

dx

1. Derivative of an integral.

2. Derivative matches upper limit of integration.

3. Lower limit of integration is a constant.

First Fundamental Theorem:

x

a

df t dt f x

dx

1. Derivative of an integral.

2. Derivative matches upper limit of integration.

3. Lower limit of integration is a constant.

New variable.

First Fundamental Theorem:

cos xd

t dtdx cos x 1. Derivative of an integral.

2. Derivative matches upper limit of integration.

3. Lower limit of integration is a constant.

sinxdt

dx

sin sind

xdx

0

sind

xdx

cos x

The long way:First Fundamental Theorem:

20

1

1+t

xddt

dx 2

1

1 x

1. Derivative of an integral.

2. Derivative matches upper limit of integration.

3. Lower limit of integration is a constant.

2

0cos

xdt dt

dx

2 2cosd

x xdx

2cos 2x x

22 cosx x

The upper limit of integration does not match the derivative, but we could use the chain rule.

53 sin

x

dt t dt

dxThe lower limit of integration is not a constant, but the upper limit is.

53 sin xdt t dt

dx

3 sinx x

We can change the sign of the integral and reverse the limits.

2

2

1

2

x

tx

ddt

dx eNeither limit of integration is a constant.

2 0

0 2

1 1

2 2

x

t tx

ddt dt

dx e e

It does not matter what constant we use!

2 2

0 0

1 1

2 2

x x

t t

ddt dt

dx e e

2 2

1 12 2

22xx

xee

(Limits are reversed.)

(Chain rule is used.)2 2

2 2

22xx

x

ee

We split the integral into two parts.

The Fundamental Theorem of Calculus, Part 2

If f is continuous at every point of , and if

F is any antiderivative of f on , then

,a b

b

af x dx F b F a

,a b

(Also called the Integral Evaluation Theorem)

We already know this!

To evaluate an integral, take the anti-derivatives and subtract.