MULTIPLE INTEGRAL

Change of Order

7 Topics
VECTOR CALCULUS

Divergence and Curl

4 Topics
Line Integral

4 Topics
Greens Theorem

6 Topics
Surface Integral

6 Topics
Gauss Divergence Theorem

8 Topics
Stokes Theorem

5 Topics
Conservative Vector Field

3 Topics
DIFFERENTIAL EQUATIONS

Orthogonal Trajectory

2 Topics
MATRICES

Introduction to Matrices

7 Topics
Linear Equations

6 Topics
Determinant

7 Topics
Revision of Matrices

6 Topics
Similar Matrices

1 Topic
DIFFERENTIAL CALCULUS

Monotonicity

5 Topics
Critical Points

3 Topics
Maxima and Minima

3 Topics
Lagrange Mean Value Theorem

5 Topics
Function of Two Variables

7 Topics
Fourier Series

1 Topic
**Example 1 :**

Change the order of integration.

**Solution :**

The given integral is

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

Let us trace all the curves to get region of integration

After changing the order of integration, the integral changes to

The domain has to be partitioned by line *y* = *a* into three regions as shown in figure.

For region *I*, x varies from to and *y* varies from *y* = 0 to *y* = *a*

For region *II*, *x* varies from to *x* = 2*a* and *y* varies from *y* = 0 to *y* =* a*

For region *III*, *x* varies from to *x* = 2*a* and *y* varies from *y* = *a* to *y* = 2*a*

So, after changing the order, the integral becomes

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