the definite integral - university of kansaspeople.ku.edu/~jila/math 125/math_125_section 5.2...
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Section 5.2The Definite Integral
(1) The Definite Integral and Net Area(2) Properties of the Definite Integral(3) Evaluating Definite Integrals using Limits
The Definite Integral of a FunctionThe definite integral of a function f on the interval [a,b] is∫ b
af (x)dx = lim
n→∞n∑
i=1f (xi )∆x
(where ∆x = b−an and xi = a+ i∆x), provided that this limit exists.
The integral symbol∫
is an elongated S , introduced by Leibniz because
an integral is a limit of sums.
The procedure of calculating an integral is integration.f (x) is the integrand and {a,b} are the limits of integration.
If∫ b
af (x)dx is defined, we say that f is integrable on [a,b].
If f is continuous on [a,b], or if f has only a finite number ofjump discontinuities, then f is integrable on [a,b].
x
y
y = f (x)
a b
+ +–∑
f (xi )∆x is an approximation of the net area.
x
y
y = f (x)
a b
+ +–∫ b
af (x)dx is the net area.
The definite integral calculates net area: the area below the positivepart of a graph minus the area above the negative part.
x
y
y = f (x)
-2 -1 1 2
1
2
3Example 1: Evaluate
∫ 2
−2f (x)dx using geometry.
Solution: The left half is a quarter-circle of radius 2and the right is a 2×2 rectangle below a base 2,height 1 triangle. The area is 5+π.
Net Area and Total Area
Example 2: For the function f (x) shown to
the right, evaluate∫ 3
−2f (x)dx .
BG
R
x
y
y= f(x)
-2 -1 1 2 3
-2
-1
1
2
Solution:∫ 3
−2f (x)dx =B +G −R = π+ 1
2−2 = π− 3
2.
In contrast, the total area contained between the graph and the x-axis is∫ 3
−2
∣∣f (x)∣∣ dx =B +G +R =π+ 52
.
Properties of the Definite Integral
Integral Property #1∫ b
af (x)dx =−
∫ a
bf (x)dx
For the first integral, the base of each rectangle in the Riemann sum haslength b−a
n , while in the second integral each rectangle has basea−bn =− b−a
n .
Integral Property #2∫ a
af (x)dx = 0
The interval [a,a] has length 0, so the region above or below the graph isjust a line segment, which has area 0.
Properties of the Definite Integral
Integral Property #3∫ b
ac dx = c(b−a)
If f (x)= c , then the area below the graph of f is a rectangle with baseb−a, height c , and therefore area c(b−a).
Integral Property #4∫ b
af (x)dx =
∫ c
af (x)dx +
∫ b
cf (x)dx
Properties of the Definite Integral
Integral Property #5∫ b
a(f (x)+g(x)) dx =
∫ b
af (x)dx +
∫ b
ag(x)dx
x
y
A
A= f (xi )∆x
a bxi xi+1
y = f (x)
x
y
B
B = g(xi )∆x
a bxi xi+1
y = g(x)
x
y
C
C = (f (xi )+g(xi ))∆x
a bxi xi+1
y = f (x)+g(x)
The brown area is the sum of the blue and the red area.
∫ b
a(f (x)−g(x)) dx =
∫ b
af (x)dx −
∫ b
ag(x)dx
Properties of the Definite Integral
Integral Property #6∫ b
a(cf (x)) dx = c
∫ b
af (x)dx for any constant c .
Stretching a graph vertically by a factor of c multiplies net area by c .
x
y
A
A= f (xi )∆x
a bxi xi+1
y = f (x)
x
y
B
B =2f (xi )∆x
a bxi xi+1
y =2f (x)
The red area is twice as the blue area.
Integral Comparison Property #1
If f (x)≥ 0 for a≤ x ≤ b, then∫ b
af (x)dx ≥ 0.
(In this case, net area just means total area under the curve.)
Integral Comparison Property #2
If f (x)≥ g(x) on the interval [a,b], then∫ b
af (x)dx ≥
∫ b
ag(x)dx .
x
y y = f (x)
y = g(x)
a b
Integral Comparison Property #3If m≤ f (x)≤M on the interval [a,b], then
m(b−a)≤∫ b
af (x)dx ≤M(b−a).
x
y
y =m
y = f (x)
y =M
a b
Evaluating Definite Integrals as LimitsThe definite integral of f on the interval [a,b] is∫ b
af (x)dx = lim
n→∞n∑
i=1f (xi )∆x
where ∆x = b−a
nand xi = a+ i∆x , provided that this limit exists.
Example 3: Express the definite integral∫ 5
1
2x1−x3 dx
as a limit of Riemann sums.
Answer: limn→∞
n∑i=1
2(1+ 4i
n
)1−
(1+ 4i
n
)3 4n
Example 4: What definite integral is represented by
limn→∞
n∑i=1
((2+ 3i
n
)sin
(2+ 3i
n
))3n? Answer:
∫ 5
2x sin(x)dx
Example 5: Evaluate the integral∫ 3
1
(x2−6
)dx .
∆x = b−a
n= 2n
xi = a+ i∆x = 1+ 2in
∫ 3
1
(x2−6
)dx = lim
n→∞n∑
i=1f (xi )∆x
= limn→∞
n∑i=1
((1+ 2i
n
)2−6
)2n
= limn→∞
n∑i=1
(8in2 + 8i2
n3 − 10n
)
= limn→∞
(8n2
n(n+1)2
+ 8n3
n(n+1)(2n+1)6
− 10nn
)= 10
3