Section 5.2 The Definite Integral (1) The Definite Integral and Net Area (2) Properties of the Definite Integral (3) Evaluating Definite Integrals using Limits
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