MATH 131-505 Spring 2015
c
Wen
Liu
5.2
5.2 The Definite Integral
Definition: If f is a function defined for a ≤ x ≤ b, we divide the interval [a, b] into n subintervals
of equal width
b−a
4x =
n
We let a = x0 , x1 , x2 , · · · , xn−1 , xn = b be the endpoints of these subintervals and we let x∗1 , x∗2 , · · · , x∗n
be any sample points in these subintervals, so x∗i lies in the ith subinterval [xi−1 , xi ]. Then the definite
integral of f from a to b is
Z b
n
X
f (x)dx = lim
f (x∗i )4x
n→∞
a
i=1
provided that this limit exists. If it does exist, we say that f is integrable on [a, b].
Note:
R
Rb
• The symbol is called an integral sign. In the notation a f (x)dx, f (x) is called the integrand and a, b are called the limits of integration; a is the lower limit and b is the upper
Rb
limit. The dx indicates that the independent variable is x. a f (x)dx is all one symbol. The
procedure of calculating an integral is called integration.
Rb
• The definite integral a f (x)dx is a number; it does not depend on x.
Z
b
b
Z
Z
a
a
• The sum
n
X
b
f (s)ds
f (t)dt =
f (x)dx =
a
f (x∗i )4x
i=1
that occurs in Definition is called a Riemann sum.
• If f takes on both positive and negative values, as in the following figure, then
Z b
f (x)dx = A1 − A2
a
where A1 is the area of the region above the x-axis and below the graph of f , and A2 is the
area of the region below the x-axis and above the graph of f .
Page 1 of 4
MATH 131-505 Spring 2015
c
Wen
Liu
5.2
Examples:
1. (p. 346) Consider f (x) = x3 − 6x. Evaluate the Riemann sum for 0 ≤ x ≤ 3 with six subintervals,
taking the sample points to be right endpoints.
2. (p. 348) Evaluate
R3
0
(x − 1)dx by interpreting it in terms of areas.
3. The graph of g consists of two straight lines and a semicircle. Use it to evaluate
R 12
3
g(x)dx.
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MATH 131-505 Spring 2015
c
Wen
Liu
5.2
If we choose x∗i to be the midpoints, we get the following approximation.
Midpoint Rule:
Z
b
f (x)dx ≈
a
n
X
f (xi )4x = 4x (f (x1 ) + · · · + f (xn ))
i=1
where
4x =
and
xi =
b−a
n
xi−1 + xi
= midpoint of [xi−1 , xi ]
2
Examples:
Z
4. (p. 349) Use the Midpoint Rule with n = 5 to approximate
1
2
1
dx.
x
5. The table gives the values of a function obtained from an experiment. Use them to estimate
R9
f (x)dx using three equal subintervals with left endpoints. If the function is known to be an
3
increasing function, can you say whether your estimate is less than or greater than the exact value
of the integral?
x
f (x)
3
4
5
6
7
8
9
-3.4 -2.2 -0.5 0.2 0.9 1.3 1.7
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MATH 131-505 Spring 2015
c
Wen
Liu
5.2
Properties of the Definite Integral
Z a
•
f (x)dx = 0
a
Z
b
a
Z
f (x)dx = −
•
f (x)dx
b
a
Z
b
•
Z
f (x)dx =
a
Z
c
Z
f (x)dx +
a
b
f (x)dx for any constant c such that a < c < b
c
b
•
cdx = c(b − a), where c is any constant
a
Z
b
•
Z
f (x)dx, where c is any constant
cf (x)dx = c
a
a
Z
b
b
Z
a
b
Z
f (x)dx ±
(f (x) ± g(x)) dx =
•
b
g(x)dx
a
a
Examples:
R 10
R8
6. If it is know that 0 f (x)dx = 17 and 0 f (x)dx = 12, find int10
8 f (x)dx.
7. Write as a single integral in the form
Rb
a
f (x)dx for
R2
f (x)dx +
−4
R5
2
f (x)dx −
R −1
−4
f (x)dx.
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