Change of Order of Integration

Thakral College Of Technology

Mathematics Presentation Submitted by

Submitted to

Ketan Tarkas

Sonendra sir

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

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

d

a

c



f(x,y)dxdy  

d

c



b

a

f(x,y)dydx

That is,



b

a

d

d

b

c

c

a

dy  f ( x, y )dx   dx  f ( x, y )dy

But if the limits are variables and the integral f(x,y) in the double integral is either difficult or even impossible integrate in the given order then we change the order of integration and corresponding change is made in the limits of integration. By geometrical considerations therefore a clear sketch of the curve is to be drawn, the new limits 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 I R: a  x  b, g1(y)  y  g2(y) Region of Type II R: c  y  d, h1(x)  x  h2(x)

Region of Type I R: a  x  b, g1(y)  y  g2(y)

Region of Type II R: c  y  d, h1(x)  x  h2(x)

For an example, let us evaluate, 2

1

  0

y/2

x2

e dxdy

A sketch of the region R is below. y

X=1

Y=2x

2 (1,2)

R 0

1

x

Since there is no elementary anti derivative of , the integral cannot be evaluated by performing the x-integration first. What we do is to evaluate this integral by expressing it as an equivalent iterated integral with the order of integration reversed. For the inside integration, y is fixed x varies from the line to the line . For the outside integration, y varies from 0 to 2.

y

(1,2)

R 0

x

This is a type 2 region or how we would describe the region is by drawing a horizontal line, tracing the left and right ylimits first and then moving the horizontal line up and down to trace the x-limits, as illustrated above. We now describe the same region as type 1 region and use the appropriate process of finding the limits

y

(1,2)

R

0

x

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

2 1

 0

y/2

1

e dxdy   e dA   x2

x2



2x

0 0

R 1

  [e y ] x

2

2x y 0

0

x2

e dxdy

1

dx   2 xe dx x2

0

 [e ]  e 1 x2 1 0

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

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

4

 0

2

y/2

f ( x, y)dxdy

Solution

The domain for the integral of the problem is labeled below. For fixed y, x is ranging from y/2 to 2 4

3

Y

X=y/2

2

1

0

X 1

2

When we reverse the order of integration we hold x fixed, with x between 0 and 2, and y ranges from 0 to 2x 4

3

Y

y=2x

2

1

0

1

X

2

Thus the reversed integral is

2

 0

2x

0

f ( x, y )dydx

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 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.

Bibliography ► Internet ► Books

Engg. Mathematics - Sonendra Gupta Engg. Mathematics - Dr. B.S. Grewal