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

d f t dt f xdx

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 d f t dt f xdx dx

x

a

d f t dt f xdx

First Fundamental Theorem:

1. Derivative of an integral.

a

xd f t dtx

f xd

2. Derivative matches upper limit of integration.

First Fundamental Theorem:

1. Derivative of an integral.

a

xd f t dt f xdx

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

d f t dt f xdx

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 dt

dx cos x 1. Derivative of an integral.

2. Derivative matches upper limit of integration.

3. Lower limit of integration is a constant.

sin xd tdx

sin sind xdx

0

sind xdx

cos x

The long way: First Fundamental Theorem:

20

1 1+txd dt

dx 2

11 x

1. Derivative of an integral.

2. Derivative matches upper limit of integration.

3. Lower limit of integration is a constant.

2

0cos

xd t dtdx

2 2cos dx 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

d t t dtdx

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

53 sin xd t t dt

dx

3 sinx x

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

2

2

1 2

x

tx

d dtdx e

Neither limit of integration is a constant.

2 0

0 2

1 1 2 2

x

t tx

d dt dtdx e e

It does not matter what constant we use!

2 2

0 0

1 1 2 2

x x

t t

d dt dtdx e e

2 2

1 12 222 xx

xee

(Limits are reversed.)

(Chain rule is used.)2 2

2 222 xx

xee

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.