DOUBLE INTEGRAL

Change of Order

7 Topics
Transformation of Variables

8 Topics
BETA AND GAMMA FUNCTION

VOLUME

SURFACE AREA

Surface Area

4 Topics
LIMITS

Introduction to Limits

4 Topics
Methods of Finding Limits

10 Topics
CONTINUITY

Continuity

4 Topics
DIFFERENTIABILITY

Differentiability

4 Topics
APPLICATION OF DERIVATIVES

Monotonicity

5 Topics
Critical Points

3 Topics
Maxima and Minima

3 Topics
MEAN VALUE THEOREM

Lagrange Mean Value Theorem

3 Topics
Part B: FUNCTION OF TWO VARIABLES

Function of Two Variables

7 Topics
INTRODUCTION

DIFFERENTIAL EQUATIONS OF FIRST ORDER AND FIRST DEGREE

ORTHOGONAL TRAJECTORY

Orthogonal Trajectory

2 Topics
DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS

CAUCHY EULER EQUATIONS

DIFFERENTIAL EQUATION OF SECOND ORDER

Section 5: LINEAR ALGEBRA

Introduction to Matrices

7 Topics
Linear Equations

6 Topics
VECTOR SPACE AND LINEAR EQUATIONS

Revision of Matrices

6 Topics
Basis and Dimensions

5 Topics
ORTHOGONALITY

Orthogonality

6 Topics
EIGENVALUES AND EIGENVECTORS

Similar Matrices

1 Topic
LINEAR TRANSFORMATION

Matrix of Linear Transformation

2 Topics
Section 6: GROUP THEORY

Cayley Table

1 Topic
Unit Group

1 Topic
Group of Matrices

1 Topic
Dihedral Groups

1 Topic
Problems on Groups

1 Topic
CYCLIC GROUP

Cyclic Groups

3 Topics
PERMUTATION GROUP

Permutation Group

5 Topics
ISOMORPHISM OF GROUPS

Isomorphism of Groups

8 Topics
EXTRNAL DIRECT PRODUCT

External Direct Product

3 Topics
HOMOMORPHISM OF GROUP

Homomorphism of Groups

8 Topics
SET THEORY

REAL NUMBERS

Real Numbers

4 Topics
TOPOLOGY ON REAL LINE

Open Sets

4 Topics
Closed Sets

1 Topic
REAL SEQUENCES

Introduction to Real Sequences

4 Topics
Limit Points

3 Topics
Subsequence

1 Topic
Cauchy Sequence

1 Topic
INFINITE SERIES

Introduction to Infinite Series

2 Topics
Alternating Series

1 Topic
POWER SERIES

Power Series

1 Topic
**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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