15 Multiple Integrals Copyright Cengage Learning All rights reserved 15 5 Applications of Double Integrals Copyright Cengage Learning All rights reserved Applications of Double Integrals Learning objectives compute double integrals in applications physics mass electric charge moments center of mass moments of inertia radii of gyration compute double integrals in applications probability with two random variables probability expected values 3 Density and Mass 4 Density and Mass We can consider a lamina with variable density Suppose the lamina occupies a region D of the xy plane and its density in units of mass per unit area at a point x y in D is given by x y where is a continuous function on D 5 Density and Mass This means that where m and A are the mass and area of a small rectangle that contains x y and the limit is taken as the dimensions of the rectangle approach 0 See Figure 1 Figure 1 6 Density and Mass To find the total mass m of the lamina we divide D into subrectangles Rij of the same size as in Figure 2 and consider x y to be 0 outside D If we choose a point in Rij then the mass of the part of the lamina that occupies Rij is approximately A where A is the area of Rij Figure 2 If we add all such masses we get an approximation to the total mass 7 Density and Mass If we now increase the number of subrectangles we obtain the total mass m of the lamina as the limiting value of the approximations 8 Density and Mass Physicists also consider other types of density that can be treated in the same manner For example if an electric charge is distributed over a region D and the charge density in units of charge per unit area is given by x y at a point x y in D then the total charge Q is given by 9 Moments and Centers of Mass 10 Moments and Centers of Mass We have found the center of mass of a lamina with constant density here we consider a lamina with variable density Suppose the lamina occupies a region D and has density function x y Recall that we defined the moment of a particle about an axis as the product of its mass and its directed distance from the axis We divide D into small rectangles 11 Moments and Centers of Mass Then the mass of Rij is approximately A so we can approximate the moment of Rij with respect to the x axis by If we now add these quantities and take the limit as the number of subrectangles becomes large we obtain the moment of the entire lamina about the x axis 12 Moments and Centers of Mass Similarly the moment about the y axis is As before we define the center of mass and so that The physical significance is that the lamina behaves as if its entire mass is concentrated at its center of mass 13 Moments and Centers of Mass Thus the lamina balances horizontally when supported at its center of mass see Figure 4 Figure 4 14 Moment of Inertia 15 Moment of Inertia Other applications of double integrals see Section 15 5 of your Stewart text moment of inertia also called the second moment of a lamina about the x or y axis or about the origin radius of gyration of a lamina with respect to the xor y axis Probability theory recall worksheet on integrating normal distribution in two variables 16 Change of Variables in Double Integrals 15 10 Copyright Cengage Learning All rights reserved 17 Change of Variables in Double Integrals Learning objective simplify double integrals by a change of variables and then evaluate a similar technique for triple integrals will be discussed later 18 Change of Variables in Double Integrals In one dimensional calculus we often use a change of variable a substitution to simplify an integral By reversing the roles of x and u we can write where x g u and a g c b g d Another way of writing Formula 1 is as follows 19 Change of Variables in Double Integrals A change of variables can also be useful in double integrals We have already seen one example of this conversion to polar coordinates The new variables r and are related to the old variables x and y by the equations x r cos y r sin and the change of variables formula can be written as f x y dA f r cos r sin r dr d where S is the region in the r plane that corresponds to the region R in the xy plane 20 Change of Variables in Double Integrals More generally we consider a change of variables that is given by a transformation T from the uv plane to the xy plane T u v x y where x and y are related to u and v by the equations x g u v y h u v or as we sometimes write x x u v y y u v 21 Change of Variables in Double Integrals We usually assume that T is a C1 transformation which means that g and h have continuous first order partial derivatives A transformation T is really just a function whose domain and range are both subsets of If T u1 v1 x1 y1 then the point x1 y1 is called the image of the point u1 v1 If no two points have the same image T is called one to one 22 Change of Variables in Double Integrals Figure 1 shows the effect of a transformation T on a region S in the uv plane T transforms S into a region R in the xy plane called the image of S consisting of the images of all points in S Figure 1 23 Change of Variables in Double Integrals If T is a one to one transformation then it has an inverse transformation T 1 from the xy plane to the uv plane and it may be possible to solve Equations 3 for u and v in terms of x and y u G x y v H x y You have seen the first examples of such transformations on the last worksheet Here is another example 24 Example A transformation is defined by the equations x u2 v2 y 2uv Find the image of the square S u v 0 u 1 0 v 1 Solution The transformation maps the boundary of S into the boundary of the image So we begin by finding the images of the sides of S 25 Example Solution continued The first side S1 is given by v 0 0 u 1 See Figure 2 Figure 2 26 Example Solution continued From the given equations we have x u2 y 0 and so 0 x 1 Thus S1 is mapped into the line segment from 0 0 to 1 0 in the xy plane The second side S2 is u 1 0 v 1 and putting u 1 in the given equations we get x 1 v2 …
View Full Document
Unlocking...