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Math 151-copyright Joe Kahlig, 13C
Sections 6.3: The Definite Integral
Definition: The definite integral of f on the interval [a, b] is
Zb
f (x)dx = lim
||P ||→0
a
n
X
f (x∗i )∆xi
i=1
b−a
, and x∗i is any value in the ith subinterval. If
n
the limit does exist, then f is called integrable on the interval [a, b].
where P is a partition of the interval [a, b], ∆xi =
If f (x) ≥ 0 on the interval [a, b], then the definite integral is the area bounded by the function f , the
x-axis, x = a and x = b.
If f (x) is not always greater than or equal to zero on the interval [a, b], then the definite integral is
the net area.
Example: Express this limit as a definite integral.
n
2 X
lim
n→∞ n
i=1
2i
3 1+
n
5
!
−6
Example: Suppose that R(t) is the rate, in gallons per hour, that water is pumped into a pool at a
water park. Explain the meaning of these integrals.
A)
R5
R(t) dt
0
B)
R4
R(t) dt
3
Example: Use the graph of f along with the indicated areas to compute these definite integrals.
A)
RA
f (x)
f (x) dx =
0
B)
RB
30
10
f (x) dx =
A
A
C)
RB
RA
A
B
C
E)
f (x) dx =
0
D)
13
RC
2f (x) dx =
A
f (x) dx =
F)
R0
A
f (x) dx =
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Math 151-copyright Joe Kahlig, 13C
Example: Compute these definite integrals.
A)
R4
1+
√
−4
B)
R3
16 − x2 dx
2x + 5 dx
0
Properties of Definite Integrals
Zb
a
Zb
a
Zb
Zb
c dx = c(b − a)
f (x) ± g(x) dx =
f (x) dx =
a
Example: If
Zc
Zb
a
f (x) dx ±
f (x) dx +
a
Z5
1
f (x) dx = 6,
Zb
Zb
g(x) dx
a
f (x) dx
a
Zb
Zb
Za
f (x) dx
a
cf (x) dx = c
a
f (x) dx = −
b
f (x) dx
c
Z10
1
g(x) dx = 10, and
Z10
1
3f (x) − 4g(x) dx = 35, then compute
Z10
5
f (x) dx.
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Math 151-copyright Joe Kahlig, 13C
Order Properties of Definite Integrals
1) If f (x) ≥ 0 for a ≤ x ≤ b, then
Zb
a
2) If f (x) ≥ g(x) for a ≤ x ≤ b, then
f (x) dx ≥ 0
Zb
a
f (x) dx ≥
Zb
a
3) If m ≤ f (x) ≤ M for a ≤ x ≤ b, then m(b − a) ≤
Z
Z
b
b
4) f (x) dx ≤
|f (x)| dx
a
a
Example: Estimate these definite integrals.
A)
Z4
ln(x) dx
Zπ
2 sin3 (x) + 1 dx
2
B)
0
g(x) dx
Zb
a
f (x) dx ≤ M (b − a)