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Area and the Definite Integral. Lesson 7.3A. a. b. The Area Under a Curve. Divide the area under the curve on the interval [a,b] into n equal segments Each "rectangle" has height f(x i ) Each width is x The area if the i th rectangle is f(x i ) • x We sum the areas. •. - PowerPoint PPT Presentation
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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