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Def. Double integral. Let z = f(x, y) be a function defined and continuous over a finite region R of the xy-plane. Let this region be subdivided into n subregions R1, R2, ... ,Rn of respective areas ΔA1, ΔA2, ... ,ΔAn. See Fig. 1. The manner of subdivision is arbitrary. In each subregion Rk we select a point Pk(xk, yk) and form the sum


Next we define the diameter ri of a subregion Ri to be the greatest distance between any two points within or on its boundary. The double integral of the function f(x, y) over the region R is then defined as  



where we stipulate that the maximum ri → 0 as n → 0.


When z = f(x, y) is non-negative over the region R the double integral 2) may be interpreted as a volume. Any term fk(xk, yk)ΔAk of 1) gives the volume of a vertical column standing on ΔAk as a base and of height zk. See Fig. 2. The sum of all these terms gives an approximation to the volume of the cylindrical column standing on a base consisting of the region R, sides generated by a line parallel to the z axis moving along the boundary of R, and top, the cut-off section of surface z = f(x, y).

Thus we see that in exact analogy to the way we defined the definite integral


as the limit of a sum and interpreted it as an area, we define the double integral as the limit of a sum and interpret it as a volume.



The iterated integral. Consider the function z = f(x, y) defined over a region R as shown in Fig. 3. The double integral of this function over region R is defined by 2) above. We will now demonstrate a method for computing this integral. We know how to find the volume of various solids by the method of slicing. The volume is expressed as the integral


where A(x) represents the area of a variable cross-section of the solid, all the sections being made by planes perpendicular to a fixed line. We shall now employ this method to evaluate the double integral.


Consider the cylindrical column shown in Fig. 4. Label the cylindrical column “Q”. Let a plane x = x', parallel to the yz-plane, be passed through column Q as shown. The cross-section of column Q created by this plane corresponds to the area under the curve z = f(x', y) between points (x', Y1) and (x', Y2), where (x', Y1) and (x', Y2) are the points where the intersecting plane crosses the boundary of region R. The area of this cross-sectional slice is given by the integral



Consider the drawing of region R shown in Fig. 5. Let a be the smallest value which x can have in R, and b the largest value. R is assumed to be a region such that the line x = x', for each value of x' between a and b, cuts the boundary of R just twice, no more. The corresponding values of y are Y1 and Y2, where Y1 < Y2. For a given value of x', the values of Y1 and Y2 can be found from the equations of the boundary of R where y = Y1(x) defines one part of the boundary and y = Y2(x) defines the other part.


Upon dropping the primes, 4) can be written as


The expression As(x) plays the role of A(x) in formula 3), representing a slice of cylinder Q. Hence the volume V of cylinder Q is given by the integral





The inner integration, with respect to y, is to be performed first. The result is a function of x, which is then integrated between the limits x = a and x = b. In the first integration x is regarded as a constant.

The expression 5) for V is called an iterated integral. It is usually written as


we have just shown the technique for evaluating double integrals. They are evaluated by an iterated integral. The relationship between iterated integrals and double integrals is so important that we state it in a theorem:

Theorem 1. The double integral of a continuous function f(x, y) over a region A of the plane can be evaluated by either of the two iterated integrals:  



where the limits of integration are determined with reference to the boundary of area A.




Def. Iterated integral. An indicated succession of integrals in which integration is to be performed first with respect of one variable, the others being held constant, then with respect to a second, the remaining ones being held constant, etc.

Indefinite iterated integrals. The integral


is an indefinite iterated integral. It is interpreted as


where the inner integral, ∫ f(x, y) dy, is integrated first, with respect to y, regarding x as a constant, and then the outer integration is performed with respect to x, regarding y as a constant. Each integration thus performed is called partial integration, and is the inverse of partial differentiation.

Example 1. Find






where the arbitrary function ole21.gif was introduced in order to provide the most general function whose partial derivative with respect to x is x3 + y3. Now performing the remaining integration with respect to y, we obtain



where Ψ(x) and Φ(y) are arbitrary functions.

One can confirm the same result will be obtained by reversing the order of integration.

The above integration is verified by observing that





Definite iterated integrals. The integral


is a definite iterated integral where a and b are constants and u1 and u2 are either constants or functions of x. The inner integral is integrated first with respect to y, regarding x as constant, and then the outer integral is integrated with respect to x, regarding y as constant.

This notation can be extended to functions of any number of variables. Thus


represents a definite iterated integral in three variables.

Example 1. Evaluate






Area computation using double integrals.


Problem. Compute the area of PQRS bounded by the curves y = f(x) and y = φ(x) and the ordinates at x = a and x = b shown in Fig. 6 using double integrals.



If we were to compute this area using single integrals we would get



Note that this integral is exactly the same as



Example. Compute the area enclosed by the parabola y2 = 2x + 4 and the line y = x - 2. See Fig. 7. 



Solution. We will integrate first with respect to x, summing the little area elements in a horizontal strip extending from the parabola to the line. The limits of integration are from x = ½ (y2 - 4) to x = y + 2. We then sum these strips by integrating with respect to y between the limits y = -2 and y = 4. Thus






Volume computation by double integration.

Problem. Compute the volume between the xy-plane and the surface z = f(x, y), bounded on the sides by two arbitrary cylindrical surfaces y = α(x) and y = β(x), and the planes x = a and x = b, as shown in Fig. 8 using double integrals


Solution. The formula is




Express the volume of one octant of the sphere x2 + y2 + z2 = r2 as a double integral.

See Fig. 9.








It is important to interpret this integration as a double summation process. The quantity z Δx Δy is the volume of a rectangular column with base Δy by Δx and height z, as indicated in Fig. 10. One can see that the first integration (with respect to y) sums these columns into a slice extending from y = 0 to y = ole46.gif The second integration, then, sums all the slices from x = 0 to x = r.



Triple integrals. 


Def. Triple integral. Let f(x, y, z) be a function defined and continuous over a finite region R of xyz-space. Suppose that R is divided into n subregions R1, R2, ... ,Rn of volumes ΔV1, ΔV2, .... , ΔVn , respectively. In each subregion Rk we select a point Pk(xk, yk, zk) and form the sum


Next we define the diameter ri of a subregion Ri to be the greatest distance between any two points within or on its boundary. The triple integral of the function f(x, y) over the region R is then defined as


where we stipulate that the maximum ri → 0 as n → 0.

Evaluation of the triple integral.

Rectangular coordinates.



where the limits of integration are chosen to cover the region R.

Cylindrical coordinates.


where the limits of integration are chosen to cover the region R.


Spherical coordinates.


where the limits of integration are chosen to cover the region R.

Centroids and moments of inertia.

Centroids. The coordinates ( ole53.gif ) of the centroid of a volume satisfy the relations




Moments of inertia. The moments of inertia of a volume with respect to the coordinate axes are given by





   Middlemiss. Differential and Integral Calculus.

   Ayres. Calculus. (Schaum) 

   Sherwood, Taylor. Calculus.

   Smith, Salkover, Justice. Calculus.

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