Thakral College Of Thakral College Of TechnologyTechnology

Mathematics Presentation

Submitted to

Sonendra sir

Submitted by

Ketan Tarkas

Change of Order of Change of Order of IntegrationIntegration

To evaluating a double integral we To evaluating a double integral we integrate first with respect to one integrate first with respect to one variable and considering the other variable and considering the other variable as constant, and then integrate variable as constant, and then integrate with respect to the remaining variable. In with respect to the remaining variable. In the former case, limits of integration are the former case, limits of integration are determined in the given region by determined in the given region by drawing stripes parallel to y-axis while in drawing stripes parallel to y-axis while in second case by drawing strips parallel to second case by drawing strips parallel to x-axisx-axis

However, if the limits are constant, the order of integration is immaterial, and in such a case we have

d

c

b

a

b

a

d

cf(x,y)dydxf(x,y)dxdy

b

a

d

c

d

c

b

adyyxfdxdxyxfdy ),(),(

That is,

But if the limits are variables and the But if the limits are variables and the integral f(x,y) in the double integral is integral f(x,y) in the double integral is either difficult or even impossible either difficult or even impossible integrate in the given order then we integrate in the given order then we change the order of integration and change the order of integration and corresponding change is made in the corresponding change is made in the limits of integration. By geometrical limits of integration. By geometrical considerations therefore a clear sketch of considerations therefore a clear sketch of the curve is to be drawn, the new limits the curve is to be drawn, the new limits are obtained.are obtained.

Reversing the order of integration

When evaluating double integrals, we can either integrate with respect to y first and then x or vice versa. In this short lesson, we will learn the method of reversing the order of integration.

Suppose we have the situation of finding the double integral of a type 2 region that is,

Region of Type IR: a x b, g1(y) y g2(y)

Region of Type IIR: c y d, h1(x) x h2(x)

Region of Type IR: a x b, g1(y) y g2(y)

Region of Type IIR: c y d, h1(x) x h2(x)

For an example, let us evaluate,For an example, let us evaluate,

A sketch of the region R is below.

2

0

1

2/

2

y

x dxdye

x

y

0 1

2

X=1 Y=2x

(1,2)

R

Since there is no elementary anti derivative Since there is no elementary anti derivative of , the integral cannot be evaluated by of , the integral cannot be evaluated by performing the performing the xx-integration first. What -integration first. What we do is to evaluate this integral by we do is to evaluate this integral by expressing it as an equivalent iterated expressing it as an equivalent iterated integral with the order of integration integral with the order of integration reversed.reversed.

For the inside integration, For the inside integration, yy is is fixed fixed xx varies from the line  to the line . varies from the line  to the line . For the outside integration, For the outside integration, yy varies from varies from 0 to 2.0 to 2.

This is a type 2 region or how we would describe the This is a type 2 region or how we would describe the region is by drawing a horizontal line, tracing the left and region is by drawing a horizontal line, tracing the left and right right yy-limits first and then moving the horizontal line up -limits first and then moving the horizontal line up and down to trace the and down to trace the xx-limits, as illustrated above.-limits, as illustrated above.

We now describe the same region as type 1 region and use We now describe the same region as type 1 region and use the appropriate process of finding the limits the appropriate process of finding the limits

x

y

0

(1,2)

R

Now that we are finding the Now that we are finding the yy limits, we limits, we just rearrange  to get , and the lower limit just rearrange  to get , and the lower limit of of yy becomes . The limits for becomes . The limits for xx easily easily follow. So,follow. So,

x

y

0

(1,2)

R

Notice how easily it is to integrate our Notice how easily it is to integrate our function  with respect to function  with respect to yy first. That is first. That is the whole point of reversing the order of the whole point of reversing the order of integration.integration.

1

0

2

0

2

0

1

2/

222 x x

R

x

y

x dxdyedAedxdye

dxxedxye xxy

x

1

0

1

0

20

22

2][

1][ 10

2

eex

Example

Sketch the region of integration and change the order of integration for

4

0

2

2/),(

ydxdyyxf

SolutionThe domain for the integral of the

problem is labeled below.For fixed y, x is ranging from y/2 to 2

X

Y

0 21

1

2

3

4

X=y/2

Y

0 21

1

2

3y=2x

When we reverse the order of integration we hold x fixed, with x between 0 and 2, and y ranges from 0 to 2x

X

4

Thus the reversed integral is

2

0

2

0),(

xdydxyxf

Why do we need to study Integration?

Often we know the relationship involving the rate of change of two variables, but we may need to know the direct relationship between the two variables. For example, we may know the velocity of an object at a particular time, but we may want to know the position of the object at that time.

The processes of integration The processes of integration are used in many applications are used in many applications

Historically, one of the first uses of integration was in finding the volumes of wine-casks (which have a curved surface).

The Petronas Towers in Kuala Lumpur experience high forces due to winds. Integration was used to design the building for strength.

The Sydney Opera House is a very unusual design based on slices out of a ball. Many differential equations (one type of integration) were solved in the design of this building.

BibliographyBibliography

► InternetInternet

►BooksBooks

Engg. Mathematics - Sonendra GuptaEngg. Mathematics - Sonendra Gupta

Engg. Mathematics - Dr. B.S. GrewalEngg. Mathematics - Dr. B.S. Grewal