Computer Algebra Systems - Calculus II - Lecture Slides, Slides of Calculus

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6.4 Integration with tables and

computer algebra systems

6.5 Approximate Integration

Tables of Integrals

x x x x C

dx u x

x

u u C

u

u du

= + + + + +

[ ]

ln( )

2 2

2 2

2

(ln ) 4 (ln ) 2 ln ln 4 (ln ) 2

1

2

2 4 (ln )^2222

2 2

a + u

• A table of 120 integrals, categorized by form, is provided on the References Pages at the back of the book.
• References to more extensive tables are given in the textbook.
• Integrals do not often occur in exactly the form listed in a table.
• Usually we need to use the Substitution rule or algebraic manipulation to transform a given integral into one of the forms in the table.

Example : Evaluate

In the table we have forms involving Let u = ln x.

Then using the integral #21 in the table,

dx x

x

2 4 (ln )

Can we integrate all continuous functions?

• Most of the functions that we have been dealing with are what are

called elementary functions.

These are the polynomials, rational functions, exponential functions,

logarithmic functions, trigonometric and inverse trigonometric

functions,

and all functions that can be obtained from these by the five operations

of addition, subtraction, multiplication, division, and composition.

• If f is an elementary function,

then f ’ is an elementary function,

but its antiderivative need not be an elementary function.

Example,

In fact, the majority of elementary functions don’t have elementary

antiderivatives.

How to find definite integrals for those functions? Approximate!

2

x

f x = e

Approximating definite integrals:

Riemann Sums

Taking more division points or subintervals in the Riemann sums, the approximation of the area of the domain under the graph of f becomes better.

Recall that the definite integral is defined as a limit of Riemann sums.

A Riemann sum for the integral of a function f over the interval [ a , b ] is

obtained by first dividing the interval [ a , b ] into subintervals and then placing a

rectangle, as shown below, over each subinterval. The corresponding

Riemann sum is the combined area of the green rectangles. The height of the

rectangle over some given subinterval is the value of the function f at some

point of the subinterval. This point can be chosen freely.

0

1

2

3

1 2 3 4

(^1 ) 1 8

y = x +

4 3

0

1

24

A = x + x

4 2 0

1 1 8

A = x + dx ∫

0 ≤ x ≤ 4

Actual area under curve:

20

3

A = (^) = 6.

→

First find the exact value

using definite integrals.

Example

0

1

2

3

1 2 3 4

(^1 ) 1 8

y = x + 0 ≤ x ≤ 4

Left endpoint

approximation:

Approximate area:

1 1 1 3 1 1 1 2 5 5. 8 2 8 4

• • • = =

(too low)

Docsity.com^ →

1 9 1 9 3 1 3 17 1 17 1 3 2 8 2 8 2 2 2 8 2 8

T

        = (^)  + (^)  + (^)  + (^)  + (^)  + (^)  + (^)  +         

1 9 9 3 3 17 17 1 3 2 8 8 2 2 8 8

T

  = (^)  + + + + + + +   

1 27

2 2

T

  = (^)    

27

4

= (^) = 6.

0

1

2

3

1 2 3 4

Averaging the areas of

the two rectangles is the

same as taking the area

of the trapezoid above

the subinterval.

Trapezoidal rule

This gives us a better approximation

than either left or right rectangles.

x a i x n

b a x

f x f x f x f x f x

x T

f x dx

n n n

b

a

= + ∆

− ∆ =

∆

≈

−

∫

i

0 1 2 1

where and

[ ( ) 2 ( ) 2 ( ) 2 ( ) ( )] 2

( )



Midpoint rule

and ( ) midpoint of [ , ]

where

( ) [ ( ) ( ) ( )]

2 1 1

1

1 2

i i i i i

n n

b

a

x x x x x

n

b a x

f x dx M x f x f x f x

= − + = −

− ∆ =

≈ = ∆ + + + ∫



Midpoint Rule: (^) 6.625 (^) (lower than the

exact value)

0.625% error

Trapezoidal Rule: (^) 6.750 (^) 1.25% error (higher than the

exact value)

Notice that the trapezoidal rule gives us an answer that

has twice as much error as the midpoint rule, but in the

opposite direction.

If we use a weighted average:

2 6.625 ( ) 6.

3

=

This is the

exact answer!

Simpson’s rule

where n is even and

2 ( ) 4 ( ) ( )]

[ ( ) 4 ( ) 2 ( ) 4 ( ) 3

( )

2 1

0 1 2 3

n

b a x

f x f x f x

f x f x f x f x

x S

f x dx

n n n

n

b

a

− ∆ =

∆

≈

− −

∫



Simpson’s rule can also be interpreted as fitting parabolas to sections of

the curve.

Simpson’s rule will usually give a very good approximation with

relatively few subintervals.

Example:

0

1

2

3

1 2 3 4

y = x +

S

( )