DEFINITE INTEGRALS 3

8/7/2019 DEFINITE INTEGRALS 3

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3.4 DEFINITE INTEGRALS

3.4.1 Definite integral and indefinite integral

The integral

is the indefinite integral of

with respect to

.

Definite integral is the integral that have limit: upper and lower limits of integration.

Suppose that then the definite integral of between and are written as

is known as definite integral of for the value of from to .

If then =

3.4.2 Properties of definite integrals

a) b) where k is a constant.

c)

The integral of a constant times a function is the constant times the integral of the

function. That is a constant can be factored out of the integrand.

d)

The integral of sum is the sum of the integrals.

e) for

For any number between and , the integral from to is the integral from to plus integral from to


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f) If for , then g)

3.4.3 Definite integration by parts

To use definite integration by parts on a definite integrals, it is important to remember

that the term on right-hand side of expression has alreadybeen integrated and so should be written in square brackets with the limits indicated.

Thus,

.

For example, to evaluate .

Put

And

Subtitute into

gives

(


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3.4.4 Definite integration by partial fractions

This process is begin just in integration of indefinite integral

To integrate rational functions, check whether the given rational function is proper or

improper. If it is improper, then use long division to convert it into proper fraction. Then,

integrate the proper rational function by express in partial fraction form.

Lastly, integrate it as definite integral that has its own limits.

For example: to evaluate .

Let

Let

Let

.

.

3.4.5 Definite integration by substitution

For definite integration by substitution, the limits of integration, originally values of, aretransformed into values of, thus allowing direct calculation of the required definite


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integral. For example, evaluate by using as thesubstitution.

Let

When When

Hence,

3.4.6 Approximate Integration

Sometimes, the integral cannot be evaluated exactly by the integrationmethod likes integration by substitution, by partial fraction or integration by part. It is

because the function may cannot be integrated algebraically. For example, .


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Since the integral gives the area bounded by the curve the x-axisand the ordinates and , an approximate value for the integration can befound by estimating this area by another method.

A common method is trapezium rule.

This method divides the area required into a number of parallel strips, each of equal

width. Each strip is regarded as a trapezium. The area of a trapezium is given by the

product of one half and the sum of its parallel sides and the perpendicular distance

between them .

Thus, if the area required is divided into n strip by (n+1) equidistant ordinates of length

y0,y1,y2,..,yn-1, yn, each strip has a width of .

Area under curve sum of areas of trapezia A1+A2+A3++An-1+An

where

Hence,

This is process known as trapezium rule. Sometimes it called Trapezoidal Rule.


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3.5 APPLICATION OF INTEGRATION

Two major application of definite integration is to find the area of a region under a

curve and to find the volume of a solid of a revolution.

3.5.1 Area Under Curve

The area bounded by the curve from and is given by

3.5.1.1 Area below the x-axis

When the graph extends below the x-axis, the corresponding value y value is negative

and so is a negative value. Thus, when an integral turns out to be negative, itindicates that the area is below the x-axis.

3.5.1.2 Area above and partly below the x-axis

Integration from x=a to x=c: gives the algebraic sum of the two parts. Hence,it is always wise to draw a sketch of the function to see whether the curve extends

below the x-axis before carrying out the integration.

The required area is then calculated by finding the positive and negative parts

separately and adding the numerical values of each part to obtain the required area.


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3.5.1.3 Area between curve and the y-axis

Integration can also be used to calculate the area between a curve and and the y-axis.

Sometimes, integral involves and not .Hence, it is necessary to express in terms of wherever it appears.

Area between the curve and the lines and is given by

Area between the curve and the lines and is given by


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Area of shaded region M+N

3.5.1.4 Area bounded by two curves

Suppose that we want to find the area of the region bounded by the curves y=f(x) and

y=g(x) and suppose that the curves intersect at x=a and x=b.

AreaAcan be treated as the difference between the two areas B and C, as shown

below.

Then,

where for


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If y1 and y2 are continuous throughout [a,b] with y1>y2, the area of the region R is

given by the formulae:


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3.5.1.5 Volume of solid revolution

Solid of revolution generated by rotating a plane about a line. The line is called axis of

rotation or axis of revolution. Cylinder, cone and sphere is some example solid

generated by revolving a plane region about x-axis or y-axis through 360 as shownbelow.

Solid Plane Region Solid generated

Cylinder

Rotate a rectangle about the x-axis

Cone

Rotate a triangle about y-axis

Sphere

Rotate a semicircle about the x-axis

a. Revolving a region about x-axis

If the curve with ordinates and rotated through a completerevolution about - axis, it will generate a solid symmetrical about , called a solidof revolution.


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Let V be the volume of the solid generated. To find V,consider a thin strip of the

original figure.

The volume generated by the strip is approximately equal to the volume generated by

the rectangle which is approximately cylindrical with radius y and thickness .Its volume is given by .If the whole plane is divided into a number of such strips, the volume of the solid is the

sum of all discs as .Total volume V

Hence the volume of solid of revolution =


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b. Revolving a region about y-axis

When the region is rotated about y-axis the volume of the rotated region is given by

In this case, need to be substituted in terms of.

3.5.1.6 Volume generated by the region between two curves

a. Region bounded by two curves about the x-axis

If for and R is the region bounded by the curve and the coordinates , and , the volume of revolution when R isrotated about is given by

b. Region bounded by two curves about the y-axis


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Volume of shaded region can be obtained from the formulae below

3.5.1.7 Volume of Revolution about the line If the region R is rotated through 360 about the horizontal line (parallel to x-axis)a solid of revolution is obtained as shown below:

When the region Rbounded by the curve the line and the coordinates and is rotated completely about the line , the volume generated isgiven by:

Volume of revolution

3.5.1.8 Volume of Revolution about the line If the shaded region is rotated through 360 about the vertical line (parallel to ) a solid of revolution is obtained as shown below:


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When the shaded region bounded by the curve the line and thecoordinates and is rotated completely about the line , the volumegenerated is given by:

Volume of revolution


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DEFINITE INTEGRALS 3

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