Definite integral

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1st Jan 1970 Mathematics Reference this

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DEFINITE INTEGRAL

Integration is an important concept in mathematics which, together with differentiation, forms one of the main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral, is defined informally to be the net signed area of the region in the xy-plane bounded by the graph of ƒ, the x-axis, and the vertical lines x = a and x = b.

APPLICATIONS OF DEFINITE INTEGRAL

Definite integrals aren’t just for area any more

  1. Any definite integral may be interpreted as a signed area.
  2. Area, volume, arc length, work, mass, fluid pressure, and accumulated financial value are

    quantities that may be calculated with definite integrals.

  3. The most important components of these problems are constructing the correct integral and

Interpreting the results.n

TWO VIEWS OF DEFINITE INTECRAL

When using the definite integral to solve various problems, it is useful to consider two different interpretations:

  1. A limit of approximating sums: The definite integral is formally defined as a limit of approximating sums using right sums.
  2. Accumulated change in an antiderivative: The Fundamental Theorem of Calculus states

where F is any antiderivative of f on [a ; b].

The difference F(b) – F(a) represents the accumulated change (or net change) in F over the

interval [a; b].

To find the accumulated change in F over [a; b], integrate f, the rate function associated with

F, over the interval [a ; b].

WHICH VIEW IS BETTER : SUM OR ANTIDERIVATIVE ?

  1. Often we need to decide which view (or interpretation) of the definite integral is the correct one for a given application. It could be that an approximating sum is acceptable or that a precise symbolic antiderivative is more appropriate.
  2. If an integral is presented in symbolic form, then antidifferentiation seems reasonable.
  3. For data given graphically or in a table, approximating sums are the logical choice.

EXPLANATION:

TRAPEZOIDAL RULE

The trapezoidal rule (also known as the trapezoid rule, or the trapezium rule in British English) is a way to approximately calculate the definite integral

The trapezoidal rule works by approximating the region under the graph of the function f(x) as a trapezoid and calculating its area. It follows that

To calculate this integral more accurately, one first splits the interval of integration [a,b] into n smaller subintervals, and then applies the trapezoidal rule on each of them. One obtains the composite trapezoidal rule:

Illustration of the composite trapezoidal rule (with a non-uniform grid)

This can alternatively be written as:

Where

(one can also use a non-uniform grid).

The trapezoidal rule is one of a family of formulas for numerical integration called Newton–Cotes formulas. Simpson’s rule is another, often more accurate, member of the same family. Simpson’s rule and other like methods can be expected to improve on the trapezoidal rule for functions which are twice continuously differentiable; however for rougher functions the trapezoidal rule is likely to prove preferable. Moreover, the trapezoidal rule tends to become extremely accurate when periodic functions are integrated over their periods, a fact best understood in connection with the Euler–Maclaurin summation formula. For non-periodic functions, however, methods with unequally spaced points such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally far more accurate; Clenshaw–Curtis quadrature can be viewed as a change of variables to express arbitrary integrals in terms of periodic integrals, at which point the trapezoidal rule can be applied accurately

SIMPSON RULE

In numerical analysis, Simpson’s rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation:

Simpson’s rule can be derived by approximating the integrand f(x) (in blue) by the quadratic interpolant P(x) (in red).

METHODS BASED ON UNDETERMINED COEFFICIENTS

NEWTON-COTES METHODS:

TRAPEZOIDAL METHOD

We have n=1 , x0 =a , x1=b and h=x1-x0.

Rn= ——(1)

Using eq 1 ,the rule can be made exact for polynomial of degree upto one.For f(x)=1 and x, we get the system of equations .

f(x)= 1: x1- x0 = + or = +

f(x) = x: ½ ( – ) = +

( ) ( ) = +

h( 2 + h ) = + ( )

h( 2 + h ) = ( + ) + h = h + h

h= , or =

From the first equation , we get h – = h /2 . The method becomes

= [ f( ) + f (]

The error constant is given by

C = – [ – ] – [ – ]

= [ 2 ( + 3 h + 3 + ) -2 -3 h -3h( + 2h + ) ]

= –

SIMPSON` S METHOD

We have n = 2 , = a , = + h , = + 2h = b , h=(b – a )/2 .We write

= f( ) + f() + f( )

The rule can be made exact for polynomials of degree upto two .

For f(x) = 1, x , , we get the following system of equations.

f(x) = 1: – = + + , or 2h = + + ———–(2)

f(x) = x: ( – ) = + + —————-(3)

f(x) = : ( – ) = + + —————–(4)

From (3) , we get

( ) ( ) = + + h) + + 2h)

(2h) (2+ 2h) = ( + + ) + ( + 2 ) h

= 2h + ( + 2 ) h

2h = + 2 ————–(5)

From (4) , we get

[( + 6 h + 12 + 8 ) – ] = + ( + 2 h + ) +

( + 4 + 2 h + ) h + )

Or h = + 4 —————(6)

Solving (5) , (6) and (2) , we obtain = , = ,

The Method is given by ………..,

= [ f() + 4 f() + f ()

The error constant is given by

C =

= –

COMPARISON BETWEEN TRAPEZOIDAL RULE AND SIMPSON’S RULE

Two widely used rules for approximating areas are the trapezoidal rule and Simpson’s rule. To motivate the new methods, we recall that rectangular rules approximated the function by a horizontal line in each interval. It is reasonable to expect that if we approximate the function more accurately inside each interval then a more efficient numerical scheme will follow. This is the idea behind the trapezoidal and Simpson’s rules. Here the trapezoidal rule approximates the function by a suitably chosen (not necessarily horizontal) line segment. The function values at the two points in the interval are used in the approximation. While Simpson’s rule uses a suitably chosen parabolic shape (see Section 4.6 of the text) and uses the function at three points.

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The Maple student package has commands trapezoid and simpson that implement these methods. The command syntax is very similar to the rectangular approximations. See the examples below. Note that an even number of subintervals is required for the simpson command and that the default number of subintervals is n=4 for both trapezoid and simpson.

> with(student):

> trapezoid(x^2,x=0..4);

> evalf(trapezoid(x^2,x=0..4));

22

> evalf(trapezoid(x^2,x=0..4,10));

21.44000000

> simpson(x^2,x=0..4);

> evalf(simpson(x^2,x=0..4));

21.33333333

> evalf(simpson(x^2,x=0..4,10));

21.33333333

EXAMPLES OF TRAPEZOIDAL AND SIMPSON`S RULE

Ques:Evaluate using trapezoidal and Simpson`s Rule with h=0.05

Sol: x0= 1 , x1= 1.05 , x2= 1.1 , x3= 1.15 , x4= 1.20 , x5=1.25 , x6= 1.3

I(trapezoidal) = = .05/2[ f(1) + 2( f (1.05) + f(1.1) +f(1.15)+ f(1.120) +f (1.25)) +f(1.3)]

= 0.326808

=

=

=

I(simpson) = [f(1) + 4 (f (1.05)+ f(1.15) + f(1.25) + 2(f(1.1) + f(1.20) +f(1.3) ]

= 0.321485

Ques 2 :Find the approximate value of

I=

Using (i) trapezoidal rule and ,(ii) Simpson`s rule.Obtain a bound for the error. The exact value of I=ln2=0.693147 correct to six decimal places.

Sol: Using the Trapezoidal rule ,

I= ( 1+ ) = 0.75

Error = 0.75 – 0.693147 = 0.056853

Using the Simpson`s Rule,

I= (1+ + ) = = 0.694444

Error = 0.694444 – 0.693147 = 0.001297

DEFINITE INTEGRAL

Integration is an important concept in mathematics which, together with differentiation, forms one of the main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral, is defined informally to be the net signed area of the region in the xy-plane bounded by the graph of ƒ, the x-axis, and the vertical lines x = a and x = b.

APPLICATIONS OF DEFINITE INTEGRAL

Definite integrals aren’t just for area any more

  1. Any definite integral may be interpreted as a signed area.
  2. Area, volume, arc length, work, mass, fluid pressure, and accumulated financial value are

    quantities that may be calculated with definite integrals.

  3. The most important components of these problems are constructing the correct integral and

Interpreting the results.n

TWO VIEWS OF DEFINITE INTECRAL

When using the definite integral to solve various problems, it is useful to consider two different interpretations:

  1. A limit of approximating sums: The definite integral is formally defined as a limit of approximating sums using right sums.
  2. Accumulated change in an antiderivative: The Fundamental Theorem of Calculus states

where F is any antiderivative of f on [a ; b].

The difference F(b) – F(a) represents the accumulated change (or net change) in F over the

interval [a; b].

To find the accumulated change in F over [a; b], integrate f, the rate function associated with

F, over the interval [a ; b].

WHICH VIEW IS BETTER : SUM OR ANTIDERIVATIVE ?

  1. Often we need to decide which view (or interpretation) of the definite integral is the correct one for a given application. It could be that an approximating sum is acceptable or that a precise symbolic antiderivative is more appropriate.
  2. If an integral is presented in symbolic form, then antidifferentiation seems reasonable.
  3. For data given graphically or in a table, approximating sums are the logical choice.

EXPLANATION:

TRAPEZOIDAL RULE

The trapezoidal rule (also known as the trapezoid rule, or the trapezium rule in British English) is a way to approximately calculate the definite integral

The trapezoidal rule works by approximating the region under the graph of the function f(x) as a trapezoid and calculating its area. It follows that

To calculate this integral more accurately, one first splits the interval of integration [a,b] into n smaller subintervals, and then applies the trapezoidal rule on each of them. One obtains the composite trapezoidal rule:

Illustration of the composite trapezoidal rule (with a non-uniform grid)

This can alternatively be written as:

Where

(one can also use a non-uniform grid).

The trapezoidal rule is one of a family of formulas for numerical integration called Newton–Cotes formulas. Simpson’s rule is another, often more accurate, member of the same family. Simpson’s rule and other like methods can be expected to improve on the trapezoidal rule for functions which are twice continuously differentiable; however for rougher functions the trapezoidal rule is likely to prove preferable. Moreover, the trapezoidal rule tends to become extremely accurate when periodic functions are integrated over their periods, a fact best understood in connection with the Euler–Maclaurin summation formula. For non-periodic functions, however, methods with unequally spaced points such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally far more accurate; Clenshaw–Curtis quadrature can be viewed as a change of variables to express arbitrary integrals in terms of periodic integrals, at which point the trapezoidal rule can be applied accurately

SIMPSON RULE

In numerical analysis, Simpson’s rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation:

Simpson’s rule can be derived by approximating the integrand f(x) (in blue) by the quadratic interpolant P(x) (in red).

METHODS BASED ON UNDETERMINED COEFFICIENTS

NEWTON-COTES METHODS:

TRAPEZOIDAL METHOD

We have n=1 , x0 =a , x1=b and h=x1-x0.

Rn= ——(1)

Using eq 1 ,the rule can be made exact for polynomial of degree upto one.For f(x)=1 and x, we get the system of equations .

f(x)= 1: x1- x0 = + or = +

f(x) = x: ½ ( – ) = +

( ) ( ) = +

h( 2 + h ) = + ( )

h( 2 + h ) = ( + ) + h = h + h

h= , or =

From the first equation , we get h – = h /2 . The method becomes

= [ f( ) + f (]

The error constant is given by

C = – [ – ] – [ – ]

= [ 2 ( + 3 h + 3 + ) -2 -3 h -3h( + 2h + ) ]

= –

SIMPSON` S METHOD

We have n = 2 , = a , = + h , = + 2h = b , h=(b – a )/2 .We write

= f( ) + f() + f( )

The rule can be made exact for polynomials of degree upto two .

For f(x) = 1, x , , we get the following system of equations.

f(x) = 1: – = + + , or 2h = + + ———–(2)

f(x) = x: ( – ) = + + —————-(3)

f(x) = : ( – ) = + + —————–(4)

From (3) , we get

( ) ( ) = + + h) + + 2h)

(2h) (2+ 2h) = ( + + ) + ( + 2 ) h

= 2h + ( + 2 ) h

2h = + 2 ————–(5)

From (4) , we get

[( + 6 h + 12 + 8 ) – ] = + ( + 2 h + ) +

( + 4 + 2 h + ) h + )

Or h = + 4 —————(6)

Solving (5) , (6) and (2) , we obtain = , = ,

The Method is given by ………..,

= [ f() + 4 f() + f ()

The error constant is given by

C =

= –

COMPARISON BETWEEN TRAPEZOIDAL RULE AND SIMPSON’S RULE

Two widely used rules for approximating areas are the trapezoidal rule and Simpson’s rule. To motivate the new methods, we recall that rectangular rules approximated the function by a horizontal line in each interval. It is reasonable to expect that if we approximate the function more accurately inside each interval then a more efficient numerical scheme will follow. This is the idea behind the trapezoidal and Simpson’s rules. Here the trapezoidal rule approximates the function by a suitably chosen (not necessarily horizontal) line segment. The function values at the two points in the interval are used in the approximation. While Simpson’s rule uses a suitably chosen parabolic shape (see Section 4.6 of the text) and uses the function at three points.

The Maple student package has commands trapezoid and simpson that implement these methods. The command syntax is very similar to the rectangular approximations. See the examples below. Note that an even number of subintervals is required for the simpson command and that the default number of subintervals is n=4 for both trapezoid and simpson.

> with(student):

> trapezoid(x^2,x=0..4);

> evalf(trapezoid(x^2,x=0..4));

22

> evalf(trapezoid(x^2,x=0..4,10));

21.44000000

> simpson(x^2,x=0..4);

> evalf(simpson(x^2,x=0..4));

21.33333333

> evalf(simpson(x^2,x=0..4,10));

21.33333333

EXAMPLES OF TRAPEZOIDAL AND SIMPSON`S RULE

Ques:Evaluate using trapezoidal and Simpson`s Rule with h=0.05

Sol: x0= 1 , x1= 1.05 , x2= 1.1 , x3= 1.15 , x4= 1.20 , x5=1.25 , x6= 1.3

I(trapezoidal) = = .05/2[ f(1) + 2( f (1.05) + f(1.1) +f(1.15)+ f(1.120) +f (1.25)) +f(1.3)]

= 0.326808

=

=

=

I(simpson) = [f(1) + 4 (f (1.05)+ f(1.15) + f(1.25) + 2(f(1.1) + f(1.20) +f(1.3) ]

= 0.321485

Ques 2 :Find the approximate value of

I=

Using (i) trapezoidal rule and ,(ii) Simpson`s rule.Obtain a bound for the error. The exact value of I=ln2=0.693147 correct to six decimal places.

Sol: Using the Trapezoidal rule ,

I= ( 1+ ) = 0.75

Error = 0.75 – 0.693147 = 0.056853

Using the Simpson`s Rule,

I= (1+ + ) = = 0.694444

Error = 0.694444 – 0.693147 = 0.001297

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