Post on 17-Jan-2016
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• 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