**Example 1 :**

Change the order of double integration

**Solution :**

The given integral is

Comparing the two integral, the curves bounding the region of integration are given by

& are the part of ellipse

The region of integration is shown in figure

After changing the order, the given integral converts to

To get the limit of *x*, let us draw an arrow parallel to *x*-axis intersecting the region of integration arbitrarily.

The arrow enters the region at *A* and leaves at point *B*.

Let us write the equation of curve at *A* and *B* in the form *x* = *g*(*y*)

At and at

The arrow has to be moved from *y* = –1 to *y* = 1 so as the traverse the whole region of integration.

Hence, the unit of *x* is from to and that of *y* from –1 to +1.

So, after changing the order

**Example 2 :**

Change the order of double integration.

**Solution :**

The given integral is

Comparing the two integral, the curves bounding the region of integration are given by

Let us draw these curves to get the region of integration.

After changing the order, the given integral converts to

To get the limit of *x*, let us draw an arrow parallel to *x*-axis intersecting the region of integration arbitrarily.

From the region of integration plotted in figure we find that if arrow is drawn below line, *y* = 2, the curves intersected are *x* = 1 and *y* = *x* but if arrow is drawn above line *y* = 2 the curves intersected are *y* = 2*x* and *x* = 2.

In this case the region of integration has to be partitioned into region I and region *II* by partition line *y* = 2.

So, the given integral will be broken into two integrals corresponding to *I* and *II* while changing the order. For region *I*, the limit of *x* varies from *x* = 1 to *x* = *y* and that of *y* varies from *y* = 1 to *y* = 2.

For region *II*, the limit of *x* varies from to *x* = 2 and that of *y* varies from *y* = 2 to *y* = 4.

So, after changing the order, the integral changes to

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