Double Integrals

Consider a function of two independent variables . Geometrically, for each in the plane, is defined, and represents a surface in the space.

Let be a region in the plane. To define the double integral of over the region , divide into finite number of sub-regions, say .

Let sub-region has area . Let be a point on the sub-region. Let . Consider . As such that maximum value of , the above sum converges to a value called the double integral of over the region denoted by .

Note that is the volume of a cylindrical solid region with the base and height .Geometrically is the volume of the cylindrical region with the base in the plane and the top surface is cut off by the surface.

If , then

Evaluation of double integrals

Let the elementary area be the area of a rectangle with sides and parallel to the co-ordinate axes.

A double integral can be considered as an iterated integral.

i.e. ,

The dependent variable among and is considered as the variable of the inner integral and the independent variable is the variable of the outer integral. While evaluating the inner integral, the variable of the outer integral is treated as a constant (i.e. the inner integration is the partial integration w.r.t the corresponding variable), then the outer integral is evaluated.

If the region is bounded by the curves , for , then .

If is bounded by the curves , for , then

Properties of the double integrals1)

If and are functions of and defined over , and are constants, then

2)

If the region is the union of two non-overlapping regions and i.e. , then

3)

If where is a function of alone and is a function of alone and if & are independent of and & are independent of , then

4)

Volume of the cylindrical region with base in the plane and top surface is cut off by the surface .5)

Area of Double integral in the polar co-ordinates

If the region in the plane is bounded by the polar curves and for , then

Problems:1. Evaluate.Solution:

or

2.

Evaluate

3.

Evaluate where

4.

Evaluate where is bounded by , and .

5.

Evaluate when

6.

Evaluate over the region bounded by and in the first quadrant.

7. Change the order of the integration and evaluate .

Changing the order of integration we get

8. Evaluate by changing the order of integration.

9. Evaluate by changing the order of integration.

10. Evaluate by reversing the order of integration.

Jacobians

Let u = u(x, y) and v = v(x, y) be continuous functions of x and y having continuous partial derivatives. Then Jacobian of u, v w.r.t. x & y denoted by J or is defined as

J = = The definition can be extended to any number of variables.

If is a function of having continuous partial derivatives , I = 1,2,n then Jacobian of w.r.t. denoted by is defined as

J = = Properties of Jacobians:1.

If , then

Proof:Let and be one-to-one and onto functions of and . Then the inverses of these functions are defined and and . Hence is defined.Consider

Using the total differential formula.

Note that if , then only is defined and .2.

If and then .

Proof:Consider

3.

If and are one-one and onto functions of and with Jacobian , then .Examples:1)

Let and .

Then .2)

Consider the transformation from the Cartesian co-ordinates to polar co-ordinates , i.e

3) Consider the transformation from Cartesian co-ordinate system to cylindrical co-ordinate system,

, a constant represents a cylinder.

, a constant represents a half-plane.

, a constant represents a plane parallel to -plane.

and represents a line.

and represents a circle.

and represents a half line (ray).

4) Consider the transformation from Cartesian co-ordinate system to spherical co-ordinate system, i.e

, a constant represents a sphere with centre at the origin and radius .

represents a half plane.

represents a cone with axis as -axis and semi-vertical angle .

and represents a semi-circle.

and represents a circle.

and represents a half line.

.Problems:1) Evaluate the following by changing to polar co-ordinates :

Solution: Changing over to polar co-ordinates , , for , we get

2)

Evaluate where is the region bounded by circles .

Solution: Changing over to polar co-ordinates , , we get 3) Evaluate by changing to polar co-ordinates.

Solution:

Changing over to polar co-ordinates , ,

4)

Evaluate using the transformations .

Solution: and

Now

Changing the variables, we get

5) Evaluate by changing over to polar co-ordinates.

Solution: Changing over to polar co-ordinates, ,

Exercises:1) Evaluate by changing over to polar co-ordinates.2)

Evaluate where is bounded by

(Hint: Use the transformation , )3)

Evaluate whereis the parallelogram in the -plane with vertices and using the transformations .4)

Evaluate where is the region bounded by one loop of .5) Evaluate by transforming into polar co-ordinates.Area enclosed by plane curves 1. Cartesian co-ordinates

Consider the area enclosed by the curves and and the ordinates and as shown in figure 1. Divide this area into vertical strips of width . If and be two neighboring points, then the area of the small rectangle is . Figure 1:

The area of the strip Since for all rectangles in the strip, is the same and varies from tothe area of the strip .Adding up all such strips from to , we get

The area .

Similarly, dividing the area (figure 2) into horizontal strips of width , we get the area .

Figure 2: Illustration:1. By double integration, find the whole area of the curve .Solution:

The required area is

Put.Then

1. Find by double integration, the area included between the curve and its asymptote.Solution: The required area is

=

1.

Find by double integration, the area included between the curve and the line .Solution: The required area is

Self-learning exercise:1. Find using double integrals, the area of a plate in the form of a quadrant of the ellipse

1. Using double integrals, find the area between the parabolas

and

Solutions:1. 1. (16/3)a2Polar Co-ordinates

Consider an area A enclosed by a curve whose equation is in polar co-ordinates. Let , be two neighbouring points. Mark circular areas of radii and meeting OQ in R and OP (produced) in S. Since arc PR=and PS=, area of the curvilinear rectangle PRQS is approximately=PR. PS=..

If the whole area is divided into such curvilinear rectangles, the sum taken for all these rectangles gives in the limit the area A. Hence A=where the limits are to be so chosen as to cover the entire area.

Illustration:1. Find the area of the cardioid , by double integration. Solution::

Area = sq.units1.

Find by double integration, the area lying inside the circle and outside the cardioid .

Solution: Area = sq.units.1.

Find the area common to the circles and by double integration.

Solution: Area = sq.units.

Exercise:1. Calculate the area included between the curve and its asymptote, using double integration.1.

Find by double integration, the area lying inside the cardioid and the circle Solutions:1.

1.

Volumes of solids1. Volumes as double integrals:

Consider a surface . Let the orthogonal projection on XY-plane of its portion be the area S. Divide S into elementary rectangles of area by drawing lines parallel to X and Y axes. With each of these rectangles as base, erect a prism having its length parallel to OZ.

volume of this prism between S and the given surface is

Hence the volume of the solid cylinder on S as base, bounded by the given surface with generators parallel to Z-axis is = where the integration is carried over the area S.

Note: While using polar co-ordinates, divide S into elements of area .. Therefore replacing by ., we get the required volume as 1. Volumes as solids of revolution

Consider an elementary area at the point of a plane area A. As this elementary area revolves about x-axis, we get a ring of volume = . Hence the total volume of the solid formed by the revolution of the area A about the x-axis =.

In polar co-ordinates, the above formula for the volume becomes .

Similarly, the volume of the solid formed by the revolution of the area A about the y-axis is 1.

Find the volume bounded by the cylinder and the planes and using a double integral.Solution

cubic units

Using a triple integral, V

1. Using a double integral, find the volume of the tetrahedron bounded by the co-ordinate planes and the plane .Solution:

cubic units

Using a triple integral, V

1. Using a double integral, find the volume generated by the revolution of the cardioid about its axis.

Solution:

cubic units