Drill: Find dy/dx
• y = -cosx
• y = sin x
• y = ln (sec x)
• y = ln (sin x)
• 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
a
b
b
a
 f ( x)dx    f ( x)dx
• Order of Integration
a
• Zero
 f ( x)dx 0
a
• Constant Multiple
b
b
a
a
 kf ( x)dx  k  f ( x)dx
b
b
a
a
  f ( x)dx    f ( x)dx
Rules for Definite Integrals
b
b
b
a
a
a
• Sum and Difference  [ f ( x)  g ( x)]dx  f ( x)dx   g ( x)dx
• Additivity
c
b
c
a
a
b
 f ( x)dx  f ( x)dx   f ( x)dx
• Max-Min Inequality: If max f and min f are the
maximum and minimum values of f on [a, b],
b
then
min f  (b  a)  f ( x)dx  max f (b  a)

a
Rules for Definite Integrals
• Domination
f(x) > g(x) on [a,b]
b
a
a
  f ( x)dx   g ( x)dx
b
f(x) > 0 on [a, b]
b
  f ( x)dx  0
a
Example 1 Using the Rules for
Definite Integrals
Suppose
3
5
4
3
 f x dx  9,  f x dx  11,
3
and  hx  dx  14.
4
Find each of the following integrals, if possible.
3
 f x dx
5
5
   f x  dx
3
 11
 11
Example 1 Using the Rules for
Definite Integrals
Suppose
3
5
4
3
 f x dx  9,  f x dx  11,
3
and  hx  dx  14.
4
Find each of the following integrals, if possible.
5
 f x  dx
4

3
5
4
3
 f x dx   f x dx
 9  11
 2
Example 1 Using the Rules for
Definite Integrals
Suppose
3
5
4
3
 f x dx  9,  f x dx  11,
3
and  hx  dx  14.
4
Find each of the following integrals, if possible.
3
3
3
4
4
4
 3 f x   4hx dx   3 f x  dx    4hx  dx
3
3
4
4
 3  f  x  dx  4  h x  dx
Example 1 Using the Rules for
Definite Integrals
3
3
3
4
4
4
dx    4h x  dx
 3 f x   4hx dx  339fx414
 29
3
3
 3  f  x  dx  4  h x  dx
4
4
Example 1 Using the Rules for
Definite Integrals
Suppose
3
5
4
3
 f x dx  9,  f x dx  11,
3
and  hx  dx  14.
4
Find each of the following integrals, if possible.
4
 f x dx
3
Not possible; not enough information given.
Example 1 Using the Rules for
Definite Integrals
Suppose
3
5
4
3
 f x dx  9,  f x dx  11,
3
and  hx  dx  14.
4
Find each of the following integrals, if possible.
6
 hx  dx
8
Not possible; not enough information given.
Example 1 Using the Rules for
Definite Integrals
Suppose
3
5
4
3
 f x dx  9,  f x dx  11,
3
and  hx  dx  14.
4
Find each of the following integrals, if possible.
5
  f x   hx dx
3
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
b
1
av f  
f x  dx.

ba a
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
1
av f  
f x  dx

ba a


5

6  x 2  2.3334
x  8.3334
2

1
2
av 6  x 
6

x
dx
x  2.887

50 0
Since 2.887 lies in the
1
interval, the function
  11.667 
5
does assume its average
 2.3334
value in the interval.
2
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
1
 sec x tan x  sec 2 x
sec x  tan x
x
x
1

 sec x(tan x  sec x)
sec x  tan x
dy / dx 
• y = ln (sec x + tan x)
dydy
/ dx

x(1x(/ ex))lne x  1
• y = xln x –x
/ dx
• y = xex
 1  ln x  1  ln x
Using Antiderivatives
for Definite Integrals
If f is integrable over the interval [a, b], then
b
 f x  dx  F b  F a 
a
where f is the derivative of F.
Determining Integrals with Power
Functions
Integrals: (where k and C are constants)
 (k )dx  kx  C
k 2
 (kx)dx  2 x  C
k 3
 (kx )dx  3 x  C
2
Note: when we are evaluating at definite
integrals, we do not need to + C.
You will need to remember your derivative rules in
order to do your anti-derivatives (integrals)
Example: If y = sin x, dy/dx = cos x
Therefore, b
b
 cos dx sin x
a
a
Example: if y = tan x, dy/dx = sec2x
b
b
Therefore,
2
 sec
xdx  tan x a
a
I would strongly suggest that you dig out your derivatives’ sheet from chapter
3! (You may use it on your next quiz!)
Example 3 Finding an Integral
Using Antiderivatives
Find each integral.
3
 3x dx  x
2
1
3
2
3 3
 cos x dx  sin x 
1
 3 1
3
 26
3
3

2
2
2
3

 sin
 sin
2
2
 1 1
 2
Example 3 Finding an Integral
Using Antiderivatives
Find each integral.
1
 e dx  e
x
1
4
x1
1
1
 e e
1
 e
e
2
e 1
 
e e
2
e 1

e
1

 sec x tan x dx

 sec x 0
0
 sec

4
4
 sec 0
 2 1