Change of Order-2

Example 1 :

Change the order of integration in

Solution :

On comparing the limits of integral is given form to its standard form

The bounding curves are given by

Let us plot all these curves

The domain of integration is the region bounded by all the four bounding curves as shown in figure.
The arrow drawn parallel to x-axis has two possibilities :

(a) If drawn below the line y = a, the arrow intersects y-axis and the parabola

(b) If drawn above the line y = a, the arrow intersects y-axis and a straight line x + y = 3a
In this case, we have to partition the domain using the partitioning line y = a into region I and region II.
The given integral will be sum of integral over region I and integral over region II.
Over region I : The limits of x is from x = 0 to and limit of y is from y = 0 to y = a

Over region II : The limit of x is from x = 0 to x = 3a – y and limit of y is from y = a to y = 3a

So, after changing the order, the integral converts to

Example 2 :

Change the order of double integration

Solution :

The region of integration is bounded by the curves

Let us plot all these curves
The region of integration is shown in figure

After changing the order, the given integral reduces to the form

Let us draw the arrow parallel to x-axis and intersecting the region of integration arbitrarily.

The arrow first intersect the curve at A. The equation of has to be written in the form x = f(y)

Since, A lies to left of x = 1. So,

The second curve it intersects is y = x which has to be written in the form i.e. x = y

So, the limit of x is from to x = y

The arrow has to be moved from y = 0 to y = 1 to cover the whole domain

So, limit of y is from y = 0 to y = 1

So, after changing the order, the integral reduces to form