• y = -cosx• y = sin x• y = ln (sec x)• y = ln (sin x)

Drill: Find dy/dx

• dy/dx = sin x• dy/dx = cos x• dy/dx = (1/sec x)(tan x sec x) = tan x• dy/dx = (1/sin x) (cos x) = cot x

Definite Integrals and Antiderivatives

Lesson 5.3

Objectives

• Students will be able to– apply rules for definite integrals and find the

average value of a function over a closed interval.

Rules for Definite Integrals

• Order of Integration

• Zero

• Constant Multiple

b

a

a

b

dxxfdxxf )()(

0)( a

a

dxxf

b

a

b

a

dxxfkdxxkf )()(

b

a

b

a

dxxfdxxf )()(

Rules for Definite Integrals

• Sum and Difference

• Additivity

• Max-Min Inequality: If max f and min f are the maximum and minimum values of f on [a, b], then

b

a

b

a

b

a

dxxgdxxfdxxgxf )()()]()([

c

b

b

a

c

a

dxxfdxxfdxxf )()()(

b

a

abfdxxfabf )(max)()(min

Rules for Definite Integrals

• Dominationf(x) > g(x) on [a,b]

f(x) > 0 on [a, b]

b

a

b

a

dxxgdxxf )()(

0)( b

a

dxxf

Example 1 Using the Rules for Definite Integrals

Suppose

Find each of the following integrals, if possible.

.14 and,11,93

4

5

3

3

4

dxxhdxxfdxxf

3

5

dxxf 5

3

dxxf

1111

Example 1 Using the Rules for Definite Integrals

Suppose

Find each of the following integrals, if possible.

.14 and,11,93

4

5

3

3

4

dxxhdxxfdxxf

5

4

dxxf

5

3

3

4

dxxfdxxf

119 2

Example 1 Using the Rules for Definite Integrals

Suppose

Find each of the following integrals, if possible.

.14 and,11,93

4

5

3

3

4

dxxhdxxfdxxf

3

4

43 dxxhxf

3

4

3

4

43 dxxhdxxf

3

4

3

4

43 dxxhdxxf

Example 1 Using the Rules for Definite Integrals

3

4

43 dxxhxf

3

4

3

4

43 dxxhdxxf

3

4

3

4

43 dxxhdxxf

14493

29

Example 1 Using the Rules for Definite Integrals

Suppose

Find each of the following integrals, if possible.

.14 and,11,93

4

5

3

3

4

dxxhdxxfdxxf

4

3

dxxf

Not possible; not enough information given.

Example 1 Using the Rules for Definite Integrals

Suppose

Find each of the following integrals, if possible.

.14 and,11,93

4

5

3

3

4

dxxhdxxfdxxf

6

8

dxxh

Not possible; not enough information given.

Example 1 Using the Rules for Definite Integrals

Suppose

Find each of the following integrals, if possible.

.14 and,11,93

4

5

3

3

4

dxxhdxxfdxxf

5

3

dxxhxf

Not possible; not enough information given.

Average (Mean) Value

If f is integrable on the interval [a, b], the function’s average (mean) value on the interval is

.1

b

a

dxxfab

fav

Example 2 Applying the Definition of Average (Mean) Value

Find the average value of f (x) = 6 – x2 on [0, 5]. Where does f take on this value in the given interval?

b

a

dxxfab

fav1

5

0

22 605

16 dxxxav

667.115

1

3334.2

3334.26 2 x

3334.82 x

887.2xSince 2.887 lies in the interval, the function does assume its average value in the interval.

Homework

• day 1: Page 290-292: 1-5 odd, 11-14, 47-49• day 2: p. 291: 19-30, 31-35 odd

Drill: Find dy/dx

• y = ln (sec x + tan x)• y = xln x –x • y = xex

)sec(tansectansec

1

sectansectansec

1/ 2

xxxxx

xxxxx

dxdy

sec(x)xx

xxxdxdy

ln1ln1

1ln)/1(/

xx eexdxdy )(/

Using Antiderivatives for Definite Integrals

If f is integrable over the interval [a, b], then

where f is the derivative of F.

aFbFdxxfb

a

Determining Integrals with Power Functions

Integrals: (where k and C are constants)

Note: when we are evaluating at definite integrals, we do not need to + C.

Ckxdxk )(

Cxk

dxkx 2

2)(

Cxk

dxkx 32

3)(

You will need to remember your derivative rules in order to do your anti-derivatives (integrals)

Example: If y = sin x, dy/dx = cos xTherefore,

Example: if y = tan x, dy/dx = sec2xTherefore,

I would strongly suggest that you dig out your derivatives’ sheet from chapter 3! (You may use it on your next quiz!)

b

a

b

a

xdx sincos

b

a

b

a

xxdx tansec2

Example 3 Finding an Integral Using Antiderivatives

Find each integral.

3

1

23 dxx3

1

3x

33 13

26

2

3

2

cos

dxx 23

2sin

x

2sin

2

3sin

112

Example 3 Finding an Integral Using Antiderivatives

Find each integral.

1

1

dxex 1

1 xe

11 ee

ee

1

4

0

tansec

dxxx 4

0sec

x

0sec4

sec

ee

e 12

e

e 12

12