First Fundamental Theorem of Calculus

Greg Kelly, Hanford High School, Richland, Washington

When we find the area under a curve by adding rectangles, the answer is called a Rieman sum.

211

8V t

subinterval

partition

The width of a rectangle is called a subinterval.

The entire interval is called the partition.

Subintervals do not all have to be the same size.

211

8V t

subinterval

partition

If the partition is denoted by P, then the length of the longest subinterval is called the norm of P and is denoted by .P

As gets smaller, the approximation for the area gets better.

P

0

1

Area limn

i iP

i

f c x

if P is a partition of the interval ,a b

0

1

limn

i iP

i

f c x

is called the definite integral of

over .f ,a b

If we use subintervals of equal length, then the length of a

subinterval is:b a

xn

The definite integral is then given by:

1

limn

in

i

f c x

1

limn

in

i

f c x

Leibnitz introduced a simpler notation for the definite integral:

1

limn b

i ani

f c x f x dx

Note that the very small change in x becomes dx.

b

af x dx

IntegrationSymbol

lower limit of integration

upper limit of integration

integrandvariable of integration

(dummy variable)

It is called a dummy variable because the answer does not depend on the variable chosen.

b

af x dx

We have the notation for integration, but we still need to learn how to evaluate the integral.

time

velocity

After 4 seconds, the object has gone 12 feet.

Let’s consider an object moving at a constant rate of 3 ft/sec.

Since rate . time = distance: 3t d

If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line.

ft3 4 sec 12 ft

sec

If the velocity varies:

11

2v t

Distance:21

4s t t

(C=0 since s=0 at t=0)

After 4 seconds:1

16 44

s

8s

1Area 1 3 4 8

2

The distance is still equal to the area under the curve!

Notice that the area is a trapezoid.

211

8v t What if:

We could split the area under the curve into a lot of thin trapezoids, and each trapezoid would behave like the large one in the previous example.

It seems reasonable that the distance will equal the area under the curve.

We can use anti-derivatives to find the area under a curve!

Fundamental Theorem of Calculus

b

af x dx F b F a

Just like we proved earlier!

Area under curve from a to x = antiderivative at x minus

antiderivative at a.

Area from x=0to x=1

Example: 2y x

Find the area under the curve from x=1 to x=2.

2 2

1x dx

2

1

31

3x

3 313

21 1

3

8 1

3 3

7

3

Area from x=0to x=2

Area under the curve from x=1 to x=2.

Example: 2y x

Find the area under the curve from x=1 to x=2.

2 , , 1, 2x x

To use your TI-83+ or TI-84+

Math

7. fnInt Enter

Function

VariableLimits of Integration

Example:

Find the area between the

x-axis and the curve

from to .

cosy x

0x 3

2x

2

3

2

3

2 2

02

cos cos x dx x dx

/ 2 3 / 2

0 / 2sin sinx x

3sin sin 0 sin sin

2 2 2

1 0 1 1

3

pos.

neg.