Multiple integral

The multiple integral is a definite integral of a function 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.Example. Let f(x, y) = 2 andExample 1. Consider the function f(x,y) = 2 sin(x) − 3y3 + 5 integrated over the domainExample 2. Consider the function f(x, y, z) = x exp(y2 + z2) and as integration region the sphere with radius 2 centered at the origin,Example 1a. The function is f(x, y) = (x − 1)2 + √y; if one adopts the substitution x′ = x − 1, y′ = y therefore x = x′ + 1, y = y′ one obtains the new function f2(x, y) = (x′)2 + √y.Example 2a. The function is f(x, y) = x + y and applying the transformation one obtainsExample 2b. The function is f(x, y) = x2 + y2, in this case one has:Example 2c. The domain is D = {x2 + y2 ≤ 4}, that is a circumference of radius 2; it's evident that the covered angle is the circle angle, so φ varies from 0 to 2π, while the crown radius varies from 0 to 2 (the crown with the inside radius null is just a circle).Example 2d. The domain is D = {x2 + y2 ≤ 9, x2 + y2 ≥ 4, y ≥ 0}, that is the circular crown in the positive y half-plane (please see the picture in the example); φ describes a plane angle while ρ varies from 2 to 3. Therefore the transformed domain will be the following rectangle:Example 2e. The function is f(x, y) = x and the domain is the same as in Example 2d. From the previous analysis of D we know the intervals of ρ (from 2 to 3) and of φ (from 0 to π). Now we change the function:Example 3a. The region is D = {x2 + y2 ≤ 9, x2 + y2 ≥ 4, 0 ≤ z ≤ 5} (that is the 'tube' whose base is the circular crown of Example 2d and whose height is 5); if the transformation is applied, this region is obtained:Example 3b. The function is f(x, y, z) = x2 + y2 + z and as integration domain this cylinder: D = {x2 + y2 ≤ 9, −5 ≤ z ≤ 5 }. The transformation of D in cylindrical coordinates is the following:Example 4a. The domain is D = x2 + y2 + z2 ≤ 16 (sphere with radius 4 and center at the origin); applying the transformation you get the regionExample 4b. D is the same region as in Example 4a and f(x, y, z) = x2 + y2 + z2 is the function to integrate. Its transformation is very easy:Example 4c. The domain D is the ball with center at the origin and radius 3a, The multiple integral is a definite integral of a function 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. Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the x-axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three-dimensional Cartesian plane where z = f(x, y) and the plane which contains its domain. If there are more variables, a multiple integral will yield hypervolumes of multidimensional functions. Multiple integration of a function in n variables: f(x1, x2, ..., xn) over a domain D is most commonly represented by nested integral signs in the reverse order of execution (the leftmost integral sign is computed last), followed by the function and integrand arguments in proper order (the integral with respect to the rightmost argument is computed last). The domain of integration is either represented symbolically for every argument over each integral sign, or is abbreviated by a variable at the rightmost integral sign: Since the concept of an antiderivative is only defined for functions of a single real variable, the usual definition of the indefinite integral does not immediately extend to the multiple integral. For n > 1, consider a so-called 'half-open' n-dimensional hyperrectangular domain T, defined as: Partition each interval [aj, bj) into a finite family Ij of non-overlapping subintervals ijα, with each subinterval closed at the left end, and open at the right end. Then the finite family of subrectangles C given by is a partition of T; that is, the subrectangles Ck are non-overlapping and their union is T. Let f : T → R be a function defined on T. Consider a partition C of T as defined above, such that C is a family of m subrectangles Cm and