Transcript e x
The Integral
chapter
6
• The Indefinite Integral
• Substitution
• The Definite Integral As a Sum
• The Definite Integral As Area
• The Definite Integral: The Fundamental
Theorem of Calculus
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Antiderivative
An antiderivative of a function f is a
function F such that
F f
Ex. An antiderivative of f ( x ) 6 x
is F ( x ) 3 x 2 2
since F ( x ) f ( x ).
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Indefinite Integral
The expression:
f ( x ) dx
read “the indefinite integral of f with respect to x,”
means to find the set of all antiderivatives of f.
Integral sign
f ( x ) dx
x is called the variable
of integration
Integrand
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Constant of Integration
Every antiderivative F of f must be of
the form F(x) = G(x) + C, where C is a
constant.
Notice
6 xdx 3 x
2
C
Represents every possible
antiderivative of 6x.
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Power Rule for the Indefinite
Integral, Part I
Ex.
x dx
n
x dx
3
x
n 1
n 1
x
C if n 1
4
C
4
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Power Rule for the Indefinite
Integral, Part II
1
x dx
1
x
dx ln x C
Indefinite Integral of ex and bx
e dx e C
x
b dx
x
x
b
x
C
ln b
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Sum and Difference Rules
f
Ex.
g dx
x x dx
2
fdx
x dx
2
gdx
xdx
x
3
3
x
2
C
2
Constant Multiple Rule
Ex.
kf ( x ) dx k
f ( x ) dx
2 x dx 2 x dx 2
3
3
x
( k constant )
4
4
C
x
4
C
2
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Integral Example/Different Variable
Ex. Find the indefinite integral:
u 7
2
3 e 2 u 6 du
u
3 e du 7
u
1
u du 2 u
3 e 7 ln u
u
2
2
du
6 du
u 6u C
3
3
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Position, Velocity, and Acceleration
Derivative Form
If s = s(t) is the position function of an
object at time t, then
Velocity = v =
ds
dt
Acceleration = a =
dv
dt
Integral Form
s (t )
v ( t ) dt
v (t )
a
(
t
)
dt
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Integration by Substitution
Method of integration related to chain
rule differentiation. If u is a function of
x, then we can use the formula
fdx
f
du / dx
du
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Integration by Substitution
Ex. Consider the integral:
3x x
2
3
5
9
dx
pick u x +5, then du 3 x dx
3
2
du
3x
9
u du
Sub to get
u
10
C
10
Integrate
x
2
3
dx
5
10
C
10
Back Substitute
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. Evaluate
x 5 x 7 dx
2
Let u 5 x 7
2
du
then
dx
10 x
x
5 x 7 dx
2
1
10 u
1/ 2
du
Pick u,
compute du
Sub in
3/2
1 u
C
10 3 / 2
5x
2
7
Integrate
3/2
C
Sub in
15
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. Evaluate
dx
x ln x 3
L et u ln x
th en xd u d x
dx
x ln x 3 u
u
3
du
2
2
C
ln x
2
2
C
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
3t
e dt
e 3t 2
Ex. Evaluate
Let u e + 2
3t
then
du
3e
3t
e dt
1
3t
dt
1
e 3 t 2 3 u du
ln u
C
3
ln e
3t
2
C
3
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Shortcuts: Integrals of
Expressions Involving ax + b
Rule
ax b
n
ax b
e
ax b
c
ax b
dx
1
dx
ax b
n 1
a ( n 1)
dx
1
C
n
1
ln ax b C
a
1
e
ax b
C
a
dx
1
c
ax b
C
a ln c
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Riemann Sum
If f is a continuous function, then the left Riemann
sum with n equal subdivisions for f over the interval
[a, b] is defined to be
n 1
f x k x
k 0
f ( x 0 ) x f ( x1 ) x ... f ( x n 1 ) x
f ( x0 )
f ( x1 ) ... f ( x n 1 ) x
where a x 0 x1 ... x n b are the
subdivisions and x ( b a ) / n.
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
The Definite Integral
If f is a continuous function, the definite integral
of f from a to b is defined to be
b
n 1
f x k x
f ( x ) dx nlim
k 0
a
The function f is called the integrand, the numbers
a and b are called the limits of integration, and the
variable x is called the variable of integration.
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Approximating the Definite
Integral
Ex. Calculate the Riemann sum for the
2
integral x dx using n = 10.
2
0
n 1
9
f x k x x k
k 0
k 0
2
1
5
2
2
2
(1 / 5) (2 / 5) ... (9 / 5) (1 / 5)
2 .2 8
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
The Definite Integral
b
f ( x ) dx
a
is read “the integral, from a to b of f(x)dx.”
Also note that the variable x is a “dummy
variable.” b
b
f ( x ) dx f (t ) dt
a
a
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
The Definite Integral As a Total
If r(x) is the rate of change of a quantity Q
(in units of Q per unit of x), then the total
or accumulated change of the quantity as x
changes from a to b is given by
b
T otal change in quantity Q
r ( x ) dx
a
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
The Definite Integral As a Total
Ex. If at time t minutes you are traveling at
a rate of v(t) feet per minute, then the total
distance traveled in feet from minute 2 to
minute 10 is given by
10
T otal change in distance
v ( t ) dt
2
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Area Under a Graph
Width: x
(n rect.)
y f ( x)
ba
n
a
b
Idea: To find the exact area under the graph
of a function.
Method: Use an infinite number of rectangles
of equal width and compute their area with a
limit.
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Approximating Area
Approximate the area under the graph of
f ( x) 2 x
2
on 0, 2
using n = 4.
A x f ( x 0 ) f ( x1 ) f ( x 2 ) f ( x 3 )
1
1
3
A f 0 f f 1 f
2
2
2
1
1
9 7
A 0 2
2
2
2 2
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Area Under a Graph
y f ( x)
a
b
f continuous, nonnegative on [a, b]. The area is
n 1
f x k x
n
A rea lim
k 0
b
f ( x ) dx
a
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Geometric Interpretation
(All Functions)
y f ( x)
R1
a
b
R3
R2
b
f ( x ) dx Area of R1 – Area of R2 + Area of R3
a
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Area Using Geometry
Ex. Use geometry to compute the integral
5
x 1 dx
1
Area =4
Area = 2
5
x 1 dx 4 2 2
1
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Fundamental Theorem of
Calculus
Let f be a continuous function on [a, b].
x
1. If A ( x )
f ( t ) dt , then A ( x ) f ( x ).
a
2. If F is any continuous antiderivative of
f and is defined on [a, b], then
b
f ( x ) dx F ( b ) F ( a )
a
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
The Fundamental Theorem
of Calculus
x
Ex. If A ( x )
3
t 5 t dt , find A ( x ).
4
a
A ( x )
3
x 5x
4
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Evaluating the Definite Integral
Ex. Calculate
5
1
5
1
1
2 x 1 dx
x
5
1
2
2 x 1 dx x ln x x 1
x
5 ln 5 5 1 ln 1 1
2
2
2 8 ln 5 2 6 .3 9 0 5 6
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Substitution for Definite Integrals
Ex. Calculate 0 2 x x
1
2
3
1/ 2
dx
let u x 3 x
du
then
dx
2x
2
1
0
2x x 3x
2
1/ 2
Notice limits change
dx
4
u
1/ 2
du
0
2
3
4
u
3/2
0
16
3
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Computing Area
Ex. Find the area enclosed by the x-axis, the
vertical lines x = 0, x = 2 and the graph of
y 2x .
2
2
0
2
3
2 x dx
0
2 x dx
3
Gives the area since 2x3 is
nonnegative on [0, 2].
1
2
2
x
4
0
Antiderivative
2 0
2
2
1
4
1
4
8
Fund. Thm. of Calculus
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.