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MATH 323 (2)

Double Integrals.pdf

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Mathematics & Statistics (Sci)
MATH 323

Double Integrals Double Integrals In calculus of a single variable the definite integral for f(x)>=0 is the area under the curve f(x) from x=a to x=b. For general f(x) the definite integral is equal to the area above the x-axis minus the area below the x-axis. The definite integral can be extended to functions of more than one variable. Consider a function of 2 variables z=f(x,y). The definite integral is denoted by where R is the region of integration in the xy-plane. For positive f(x,y), the definite integral is equal to the volume under the surface z=f(x,y) and above xy-plane for x and y in the region R. This is shown in the figure below. For general f(x,y), the definite integral is equal to the volume above the xy-plane minus the volume below the xy-plane. This page includes the following sections: Applications Brief Discussion of Riemann Sums Double Integrals over Rectangular Regions Example Double Integrals over General Regions Example Applications Double integrals arise in a number of areas of science and engineering, including computations of Double Integrals Area of a 2D region Volume Mass of 2D plates Force on a 2D plate Average of a function Center of Mass and Moment of Inertia Surface Area Brief Discussion of Riemann Sums As in the case of an integral of a function of one variable, a double integral is defined as a limit of a Riemann sum. Suppose we subdivide the region R into subrectangles as in the figure below (say there are M rectangles in the x direction and N rectangles in the y direction). Label the rectangles R_ij where 1<=i<=M and 1<=j<=N. Think of the definite integral as representing volume. The volume under the surface above rectangle R_ij is approximately f(x_i,y_j)A_ij, where A_ij is area of the rectangle and f(x_i,y_j) is the approximate height of the surface in the rectangle. Here (x_i,y_j) is some point in the rectangle R_ij. If we sum over all rectangles we have In the limit as the size of the rectangles goes to 0, the sum on the right converges to a value which is the definite integral. The quantity f(x,y)dAin the definite integral represents the volume in some infinitesimal region around the point (x,y). The region is so small that the f(x,y) only varies infinitesimally in the region. The double integral sign says: add up volumes in all the small regions in R. Double Integrals over a Rectangular Region Suppose that f(x,y) is continuous on a rectangular region in the xy plane as shown above. The double integral Double Integrals represents the volume under the surface. We can compute the volume by slicing the three-dimensional region like a loaf of bread. Suppose the slices are parallel to the y-axis. An example of slice between x and x+dx is shown in the figure. In the limit of infinitesimal thickness dx, the volume of the slice is the product of the cross-sectional area and the thickness dx. The cross sectional area is the area under the curve f(x,y) for fixed x and y varying between c and d. (Note that if the thickness dx is infinitesimal, x varies only infinitesimally on the slice. We can assume that x is constant.) The picture below shows the cross-sectional area. The area is given by the integral Double Integrals The variable of integation is y and x is a CONSTANT. The cross-sectional area depends on x and this is why we write C=C(x). The volume of the slice between x and x+dx is C(x)dx. The total volume is the sum of the volumes of all the slices between x=a and x=b: If substitute for C(x), we obtain: This is an example of an iterated integral. One integrates with respect to y first, then x. The
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