Riemann sums & definite integrals (4.3)

Riemann sums & definite integrals (4.3)

January 30th, 2013January 30th, 2013

I. Riemann SumsDef. of a Riemann Sum: Let f be defined on the closed interval [a, b], and let be a partition of [a, b] given by where is the width of the ith subinterval. If is any point in the ith subinterval, then the sum

is called the Riemann Sum of f for the partition .

Δa=x0 < x1 < x2 < ... < xn−1 < xn =bΔxi ci

f (ci )Δxii=1

n

∑ , xi−1 ≤ci ≤xi

Δ

*The norm of the partition is the largest subinterval and is denoted by . If the partition is regular (all the subintervals are of equal width), the norm is given by .

ΔΔ

Δ =Δx =b − a

n

x1 x2 x3

Δ

II. definite integralsDef. of a Definite Integral: If f is defined on the closed interval [a, b] and the limit

exists, then f is integrable on [a, b] and the limit is denoted by

This is called the definite integral.

limΔ → 0

f(ci )Δxii=1

n

∑

limΔ → 0

f(ci )Δxi = f(x)dxa

b

∫i=1

n

∑upper limit

lower limit

*An indefinite integral is a family of functions, as seen in section 4.1. A definite integral is a number value.

f (x)dx∫

f (x)dxa

b

∫

Thm. 4.4: Continuity Implies Integrability: If a function f is continuous on the closed interval [a, b], then f is integrable on [a, b].

Ex. 1: Evaluate by the limit definition.

xdx−2

3

∫

Thm. 4.5: The Definite Integral as the Area of a Region: If f is continuous and nonnegative on the closed interval [a, b], then the area of the region bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b is given by Area = .

f (x)dxa

b

∫

Ex. 2: Sketch a region that corresponds to each definite integral. Then evaluate the integral using a geometric formula.

a.

b.

c.

3dx−2

2

∫

x

2dx

1

4

∫

(a−x−a

a

∫ )dx

III. properties of definite integralsDefs. of Two Special Definite Integrals:

1. If f is defined at x = a, then we define .

2. If f is integrable on [a, b], then we define .

f (x)dx =0a

a

∫

f (x)dx =− f(x)dxa

b

∫b

a

∫

Thm. 4.6: Additive Interval Property: If f is integrable on the three closed intervals determined by a, b, and c, where a<c<b, then .

f (x)dx = f(x)dx+ f(x)dxc

b

∫a

c

∫a

b

∫

Thm. 4.7: Properties of Definite Integrals: If f and g are integrable on [a, b] and k is a constant, then the functions of and are integrable on [a, b], and1.

2.

kf f ±g

kf (x)dx =k f (x)dxa

b

∫a

b

∫

[ f (x)±g(x)]dx= f (x)dx± g(x)dxa

b

∫a

b

∫a

b

∫Thm. 4.8: Preservation of Inequality: 1. If f is integrable and nonnegative on the closed interval [a, b], then .

2. If f and g are integrable on the closed interval [a, b] and for every x in [a, b], then .

0≤ f(x)dxa

b

∫

f (x)dx ≤ g(x)dxa

b

∫a

b

∫

f (x)≤g(x)

Ex. 3:Evaluate each definite integral using the following values.

a.

b.

c.

d.

x3 dx0

2

∫ =4, x3 dx =60,2

4

∫ xdx =80

4

∫ , dx =40

4

∫

x3 dx0

4

∫

xdx4

4

∫

x3 −4x+9( )dx0

4

∫

xdx4

0

∫