1.0
Introduction
The multiple integral is a generalization of the definite integral to functions of more than one real variable, for example, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region
in R2 are called double integrals, and integrals of a function of three variables over a region of R3 are called triple integrals.
Definition of Double Integral The definite integral can be extended to functions of more than one variable. Mainly to determine interval using horizontal and vertical strip which both produce same answer. Consider, for example, a function of two variables z = f (x,y). The double integral
Where R is the region of integration in the xy-plane. If the definite integral
of a
function of one variablef (x) ≥ 0 is the area under the curve f (x) from x = a to x = b, then the double integral is equal to the volume under the surface z = f (x,y) and above the xy-plane in the region of integration R .
Definition of the Triple Integral We have seen that the geometry of a double integral involves cutting the two dimensional region into tiny rectangles, multiplying the areas of the rectangles by the value of the function there, adding the areas up, and taking a limit as the size of the rectangles approaches zero. We have also seen that this is equivalent to finding the double iterated iterated integral. We will now take this idea to the next dimension. Instead of a region in the xy-plane, we will consider a solid in xyz-space. Instead of cutting up the region into rectangles, we will cut up the solid into rectangular solids. And instead of multiplying the function value by the area of the rectangle, we will multiply the function value by the volume of the rectangular solid. 3 coordinates system which we need to know cartesian coordinate, cylindrical coordinate and spherical coordinate.
1
We can define the triple integral as the limit of the sum of the product of the function times the volume of the rectangular solids. Instead of the double integral being equivalent to the double iterated integral, the triple integral is equivalent to the triple iterated integral.
2
2.0
Theory
Area by Double Integration
Double Integrals in Polar Form
Triple Integrals in Rectangular Coordinates
Triple Integrals in Cylindrical and Spherical Coordinates
3
4
3.0
Question
Find the volume of the tetrahedron bounded by the planes x + y + z = 5, x = 0, y = 0, z = 0
4.0
Fila Table
FACTS (F) -Theorem of multiple integral’s rule to find the value of equation.
IDEAS (I)
LEARNING ISSUES (L)
-Integrate the equations and find the answer. Substitute the integration value into the equation.
-What is the usage of the triple integral?
-Solve the equation.
-How to find the solution?
-Which types of integrals are used?
ACTIONS (A) -Internet searching -Module referring
5
5.0
Solution
Find the volume of the tetrahedron bounded by the planes x + y + z = 5, x = 0, y = 0, z = 0 The equation of the plane x + y + z = 5 can be rewritten in the form
By setting z = 0, we get
Fig.4
Fig.5
Hence, the region of integration D in the xy-plane is bounded by the straight line y = 5 − x as shown in Figure 5.
Representing the triple integral as an iterated integral, we can find the volume of the tetrahedron: 6
V=
V=
V=
V=
V=
V=
V=
V=
V=
V=
V=
V=
7
6.0
Conclusion
In this chapter, we saw that an ''integral of an integral'' is known as an iterated integral, whereas a double integral is the limit of double sums over ever finer partitions. However, if a region R is a type I region, then =
where x = a, x = b, y = p( x) , and y = q( x) are the boundaries of R. Similarly, a double integral over a type II region can be reduced to a type II iterated integral, and in both cases, the double integral yields the volume of the solid between z = f( x,y) and the xy-plane over the region R when f( x,y) 0. Other applications of the double integral include the area of a region, the mass and center of mass of a laminate, and the computation of probabilities for joint density functions. Besides type I and type II regions, double integrals can be reduced to iterated integrals over regions in other coordinate systems. In that case, the differential is multiplied by the absolute value of the Jacobian determinant. Among the most important of these coordinate systems are polar coordinates, and indeed, the calculation of double integrals in polar coordinates is important in many applications of statistics. Finally, the double integral concept can be extended to three or more integrals. Triple integrals often occur in association with densities, where a density is the measure of the amount of a physical quantity per unit volume of a geometric solid. Triple integrals in applications also occur frequently in either cylindrical or spherical coordinates, particularly when those applications involve regular solids such as spheres and right circular cylinders. 8
7.0
References
1) 2) 3) 4)
http://en.wikipedia.org/wiki/Multiple_integral http://tutorial.math.lamar.edu/Classes/CalcIII/MultipleIntegralsIntro.aspx https://www.math.hmc.edu/calculus/tutorials/multipleintegration/ Module Engineering mathematics 3, UTHM.
9
