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