antidifferentiation: the indefinite intergral chapter five

Antidifferentiation: The Indefinite Intergral

Chapter Five

§5.1 Antidifferetiation

§5.1 General Antiderivative of a Function

§5.1 General Antiderivative of a Function

§5.1 Rules for Integrating Common Function

The Constant Rule

§5.1 Rules for Integrating Common Function

Example:

Solution:

§5.1 Applied Initial Value Problems

An initial Value problems is a problem that involves solving a differential equation subject to a specified initial condition. For instance, we were required to find y=f(x) so that

A Differential equation is an equation that involves differentials or derivatives.

We solved this initial problem by finding the antiderivative

And using the initial condition to evaluate C.

The population p(t) of a bacterial colony t hours after observation begins is found to be change at the rate

If the population was 2000,000 bacteria when observations began, what will be population 12 hours later?

Example:

Solution:

§5.2 Integration by Substitution

How to do the following integral?

§5.2 Integration by Substitution

Think of u=u(x) as a change of variable whose differential is

Then

Example:

Solution:

Find

Example:

Solution:

Example:

Solution:

To be continued

Example:

Solution:

Example:

Solution:

§5.3 The Definite Integral and the Fundamental Theorem of Calculus

All rectangles have same width.

• n subintervals:

• Subinterval width

•Formula for xi:

• Choice of n evaluation points

Right-endpoint approximation

left-endpoint approximation

Midpoint Approximation

Example:

=0.285

To be continued

=0.3325

=0.385

Example:

left-endpoint approximation

Midpoint Approximation

Right-endpoint approximation

00=1.098608585 =1.098611363

Area Under a Curve Let f(x) be continuous and satisfy f(x)≥0on the interval a≤x≤b. Then the region under the curve y=f(x)over the interval a≤x≤b has area

1 21

lim lim[ ( ) ( ) ... ( )] lim ( )n

n n jn n n

j

A S f x f x f x x f x x

Where xj is the point chosen from the jth subinterval if the Interval a≤x≤b is divided into n equal parts, each of length

b ax

n

§5.3 The Definite Integral

Riemann sum Let f(x) be a function that is continuous onthe interval a≤x≤b. Subdivide the interval a≤x≤b into n equal

parts, each of width ,and choose a number xk from the

kth subinterval for k=1, 2, …, . Form the sum

b ax

n

Called a Riemann sum.

Note: f(x)≥0 is not required

§5.3 The Definite Integral

The Definite Integral the definite integral of f on the interval

a≤x≤b, denoted by , is the limit of the Riemann sum asn→+∞; that is

b

af(x)dx

The function f(x) is called the integrand, and the numbers a and b are called the lower and upper limits of integration, respectively. The process of finding a definite integral is called definite integration.

Note: if f(x) is continuous on a≤x≤b, the limit used to define integral exist and is same regardless of how the subinterval representatives xk are chosen.

b

af(x)dx

§5.3 Area as Definite Integral

If f(x) is continuous and f(x)≥0 for all x in [a,b],then

( ) 0b

af x dx

and equals the area of the region bounded by the graph f and the x-axis between x=a and x=b

If f(x) is continuous and f(x)≤0 for all x in [a,b],then

( ) 0b

af x dx

And equals the area of the region bounded by the graph f and the x-axis between x=a and x=b

( )b

af x dx

§5.3 Area as Definite Integral

equals the difference between the area under the graph

of f above the x-axis and the area above the graph of f below the x-axis between x=a and x=b

This is the net area of the region bounded by the graph of f and the x-axis between x=a and x=b

( )b

af x dx

§5.3 The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus If the function f(x) is continuous on the interval a≤x≤b, then

( ) ( ) ( )b

af x dx F b F a

Where F(x) is any antiderivative of f(x) on a≤x≤b

Another notation:

( ) ( ) | ( ) ( )b b

aaf x dx F x F b F a

§5.3 The Fundamental Theorem of Calculus (Area justification )

In the case of f(X)≥0, represents the area the curve y=f(x) over the interval [a,b]. For fixed x between a and b let A(x) denote the area under y=f(x) over the interval [a,x].

( )b f x dxa

By the definition of the derivative,

Differentiation

Indefinite Integration

Definite integration

Example

§5.3 Integration Rule

Subdivision Rule

§5.3 Subdivision Rule

Example

Solution:

Example

Solution:

To be continued

§5.3 Substituting in a definite integral

23

3

222 3

300

1 1 2 21

3 3 31

2 41

3 31

xdx du u x

ux

xdx x

x

2.

§5.3 Substituting in a definite integral

Example

Solution:

Example

Solution: