Area and the Definite Integral

Lesson 7.3A

The Area Under a Curve

Divide the area underthe curve on the interval [a,b] inton equal segments• Each "rectangle" has height f(xi)

• Each width is x

• The area if the i th rectangle is f(xi)•x

• We sum the areas

2

a bx

•ix

1

( )n

ii

A f x x

The SumCalculated

Consider the function2x2 – 7x + 5

Use x = 0.1

Let the = left edgeof each subinterval

Note the sum

3

x 2x 2̂-7x+5 dx * f(x)4 9 0.9

4.1 9.92 0.9924.2 10.88 1.0884.3 11.88 1.1884.4 12.92 1.2924.5 14 1.44.6 15.12 1.5124.7 16.28 1.6284.8 17.48 1.7484.9 18.72 1.872

5 20 25.1 21.32 2.1325.2 22.68 2.2685.3 24.08 2.4085.4 25.52 2.5525.5 27 2.75.6 28.52 2.8525.7 30.08 3.0085.8 31.68 3.1685.9 33.32 3.332

Sum = 40.04

kx

The Area Under a Curve

The accuracy of the summation will increase if we have more segments• As we increase n

As n gets infinitely large the summation is exact

4

1

lim ( )n

in

i

A f x x

The Definite Integral

We will use another notation to represent the limit of the summation

5

1

( ) limnb

ka nk

A f x dx f x x

The integrandThe integrand

Upper limit of integration

Upper limit of integration

Lower limit of integration

Lower limit of integration

Example

Try

Use summation on calculator.

6

3 4

24

11

use (1 )k

x dx S f k x x

b ax

n

Example

Note increased accuracy with smaller x

7

Limit of the Sum

The definite integral is the limit of the sum.

8

3

2

1

x dx

Practice

Try this

What is the summation?

• Where

Which gives us

Now take limit 9

2 2

0 0

2 ( )xdx f x dx

2 0 2x

n n

1

(0 )n

k

f k x x

Practice

Try this one

• What is x?

• What is the summation?

For n = 50?

Now take limit 10

4

2

0

2 x dx

4 0 4

n n

1

(0 )n

k

f k x x

Assignment

Lesson 7.3A

Page 458

Exercises 6 – 20 all

11