The Definite Integral
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Transcript of The Definite Integral
The Definite IntegralSection 14.3
Definite integral
• As the number of integrals increase while doing the Riemann sum, the answer becomes more accurate. The limit of the Riemann Sum is called the definite integral of f from a to b, written:
b
a
dxxf )(
Example 1
• Use integral notation to express the area of the region bounded by the x-axis, the graph of g(x) = 5x5 – 3x4 and the lines x = 10 and x = 25
25
10
45 35 dxxx
Example 2
• Find the exact value of
Draw a picture!
dxx 12
3
256
Trapezoid with A = ½ (b1 + b2)h
• A = ½ (f(3) + f(12)) 9∙• f(12) = 97, f(3) = 43 630
The Anti-derivative
• This is exactly the opposite of the derivative. We have to ask ourselves, what number will give us this derivative.
x3 2
2
3x
Try some others!
a.
b.
47 x xx 42
7 2
523 xxxxx 5
3
1
4
1 34
Once we find the anti-derivative..
Evaluate it at the upper and lower bound. Then, subtract!
Back to example 2!
• Find the exact value of
dxx 12
3
256 xx 253 2
123
2 |253 xx 732 102 630
Example 3
• Find the exact value of
dx
8
10
7 x7
810|7
x 56 70 14
Example 4
• Calculate:
• This one is a little harder to integrate, so draw a picture!
dxx 10
0
21005
Example 4
¼ (10 * 50) π125 π
x7
Homework
Pages 831 – 8323 – 14
#10 is extra credit