Multiple Integrals

Multiple Integrals

Double integrals

• Double integrals over rectangular regions

,

• Double integrals over general regions

Enclose D in a large rectangle R and define.

• Area interpretation

If , then each term in the sum yields an area. Since each of the terms is

positive the sum represents the actual area and not the net, signed or accumulated

area.

• Volume interpretation

If for all , then the double integral represents the volume of

the region above and below the surface . If can be both positive

and negative then the double integral represents the (volume above the x-y plane) -

(volume below the x-y plane) or the net signed volume.

• Interpreting the integrand as a density function

Suppose that is the density of a thin metal plate. Then the double integral of

represents the total mass of the plate.

• Average value

• Properties

.

Iterated integrals

• Iterated integrals

The double integrals over some simple regions may be evaluated by computing

two successive single integrals. This two stage integration process is called iterated

integration. The process is implemented by first integrating with respect to y

(holding x constant) and then integrating with respect to x or vice versa. The outer

limits of integration must be constant. The inner limits of integration should be

constants or functions of the outer variable.

• Rectangular regions

• Type I regions

,

• Type II regions

,

• Reversing the order of integration

Some regions in the plane may be expressed as Type I or Type II regions. For these

regions the order of integration may be reversed to make the integration easier.

Convert to or vice versa.

Triple integrals

• Triple integrals over rectangular solids

,

• Volume interpretation

If , then each term in the sum yields an volume. Since each of the

terms is positive the sum represents the actual volume and not the net, signed or

accumulated volume.

• Interpreting the integrand as a density function

Suppose that is the density of a solid. Then the triple integral of

represents the total mass of the solid.

• Average value

• Properties

See properties of double integrals.

• Evaluation

We can use iterated integration provided that the region is simple in one of the 6

possible orders of integration.

By simple we mean that (i) the outer limits of integration must be constant (ii) the

middle limits of integration should be constants or functions of the outer variable (iii)

the inner limits of integration should be constants or involve the two outer variables.

For example if the region is simple in y then

If the region of integration cannot be broken into simple regions then we must use

approximate numerical solutions.

Double integrals in polar coordinates

• Rationale

Set up the double in polar coordinates if (i) the function is simpler in polar coordinates

or (ii) the region of integration is easier to describe in polar coordinates.

• Area of a polar rectangle

Recall the area of a sector is . The area of a polar rectangle is the area of

the larger sector minus the area of the smaller sector and is computed to be

• Double integral over a polar rectangle

,

• Evaluation

Use iterated integration with .

Polar rectangles

Type I regions

,

Type II regions

,

Triple integrals in cylindrical coordinates

• Cylindrical coordinates

Cylindrical to rectangular

Rectangular to cylindrical

where the quadrant of is determined by the signs of x and y.

• Volume of a cylindrical wedge

The volume of a cylindrical wedge is computed by multiplying the area of a polar

rectangle by the height to obtain

• Triple integral over cylindrical wedge

• Evaluation

Use iterated integration with provided that the region is simple in

one of the 6 possible orders of integration.

For example if the region is simple in then

If the region of integration cannot be broken into simple regions then we must use

approximate numerical solutions.

Triple integrals in spherical coordinates

• Spherical coordinates

Spherical to rectangular

Rectangular to spherical

where the quadrant of is determined by the signs of x and y.

Comment: The range of is and includes the allowable range of .

Thus we may use without any special thought. The range of is

does not correspond to the allowable range of given by .

Hence we must use the signs of x and y to determine the quadrant of .

• Volume of a spherical wedge

In order to express the volume element in spherical coordinates we note that

the volume element can be approximated by a box with curved sides. One edge of the

box has length . The edge parallel to the x-y plane is the arc of a circle with radius

and angle and has length . The remaining side comes from

rotating the radius through an angle and has length . Hence the volume

• Triple integral over a spherical wedge

• Evaluation

Use iterated integration with provided that the region is

simple in one of the 6 possible orders of integration.

For example if the region is simple in then

If the region of integration cannot be broken into simple regions then we must use

approximate numerical solutions.

Applications

• Probability density functions

A function p(x,y) is called a joint probability density function if for all

x and y and .

• Center of mass

The motion of a solid object can be analyzed by thinking of the mass as being

concentrated at a single point called the center of mass. The center of mass

of a continuously distributed mass along a line is

.

The center of mass of a continuously distributed mass in the plane is

.

The center of mass of a continuously distributed solid is

.

• Moments of inertia

Mass is a measure of the tendency of matter to resist a change in translational motion.

The moment of inertia is a measure of the tendency of matter to resist a change in its

rotational motion. The moments of inertia of a laminar region about the x and y axes

are given by

The moments of inertia of a solid about the x,y and z axes are given by

Jacobians

Jacobians are used to implement a change of variables in multiple integrals.

Theorem: If the one-to one change of variables

maps the region R in the u-v plane into the region A in the x-y plane and the Jacobian

is nonzero and never changes sign then

.

The theorem demonstrates there are three steps necessary to implement a change of variables in double integrals.

• Convert the integrand from a function of x and y to a function of u and v.
• Convert the region A in the x-y plane to a region R in the u-v plane
• Convert the area element to the new variables u and v by introducing the absolute value of the Jacobian .

There is a similar change of variable formula for triple integrals