fundamental theorems of calculus 6.4. the first (second?) fundamental theorem of calculus if f is...

Fundamental Theorems of Calculus

6.4

The First (second?) Fundamental Theorem of Calculus

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.

Second 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:

Example Applying the Fundamental Theorem

Find sin .xdtdt

dx

sin sinxdtdt x

dx

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.

Example Variable Lower Limits of Integration

5Find if sin .x

dyy t tdt

dx

2

1cos

xdtty

2

2

1

2

1

cos2

2)cos(

cos

cos

cos

xx

xx

dx

duu

dx

dudtt

du

ddx

du

du

dy

dx

dy

xuanddtty

u

u

Example The Fundamental Theorem with the Chain Rule

2

1Find / if sin .xdy dx y tdt

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.

Second (first?) FTOC

b

aaFbFdxxf )()()(

3

1

3 )1( dxx

24

14

13

4

81

4

3

1

4

xx

)(0)(

))()(()(

)()(

)()(

xfxf

aFxFDdttfD

aFxF

tFdttf

x

a xx

x

a

x

a

How to Find Total Area Analytically

To find the area between the graph of ( ) and the -axis over the interval

[ , ] analytically,

1. partition [ , ] with the zeros of ,

2. integrate over each subinterval,

3. add the absolute values o

y f x x

a b

a b f

f

f the integrals.

How to Find Total Area Analytically