VIEWS: 20 PAGES: 11 POSTED ON: 3/24/2011 Public Domain
Calculus 3 14 Multiple Integration Homework Problems Section 14.1 1, 6, 11, 16, 21, 27, 31, 37, 45, 53, 58, 59, 63, 67, 69, 73, 77, 79 Section 14.2 3, 5, 9, 14, 19, 25, 30, 35, 40, 41, 49, 55, 58, 65, 68, 71, 74, 76, 80, 83 Section 14.3 1 –8, 11, 17, 21, 25, 31, 39, 45, 49, 51, 53, 58 Section 14.4 3, 7, 12, 17, 22, 29, 49, 54, 57 Section 14.5 2, 7, 12, 17, 25, 27, 31, 35, 40 Section 14.6 5, 9, 14, 21, 25, 30, 31, 35, 37, 41, 55, 57, 61, 63, 71 Section 14.7 4, 11, 16, 19, 24, 29, 33, 35, 37, 42, 46 Section 14.8 2, 7, 9, 13, 18, 22, 23, 25, 27, 30, 31 1 Topics Integration Iterated integrals Integrals of two variables Riemann sums Volume Su¢ cient condition for integrability Arithmetic properties Evaluating double integrals Integrating with polar coordinates Riemann sums for polar sectors Center of mass and moments of inertia Mass Moment of inertia Surface area Triple integrals General change of coordinates Transformation of a rectangle Jacobian s Review: Simpson’ rule Review: Wallis’formula 2 Integration The primary focus of our e¤orts to this point in time has been to discuss di¤erentiation, and in this chapter we will concern ourselves with integration. To some extent, just like the notion of partial derivative, it is rather easy to progress in a certain direction. The concept of iterated integral, which is the standard approach to discussing integration involving more than one variable, is the pure analogue to partial di¤erentiation. Iterated integrals With an iterated integral, we hold all variables …xed but one for each part of the iteration. For a function f (x; y) we can integrate, for example, by holding y constant and integrating with respect to x: Z b f (x; y) dx (1) a It is easy to give examples using polynomials. Because y is being treated as a constant, we can allow the limits of integration a, and b to be functions of y: a = a (y) (2) b = b (y) (3) Then we can integrate over y to obtain an iterated integral: Z d Z b(y) f (x; y) dxdy (4) c a(y) Similarly, we can have an iterated integral in which y is integrated …rst, with x held …xed, and then we integrate over x: Z m Z h(x) f (x; y) dydx (5) k g(x) Together, the two integrations determine a region of integration in the xy- plane. For example, the region associated with (4) is c < y < d and, for a …xed y, a (y) < x < b (y) (6) Due to this, we can interpret the case when f (x; y) = 1 as giving the area between two curves in the plane. Actually, sometimes we can reverse the order of integration to …nd two di¤erent ways of representing the area. Integrals of two variables Section 14.2 starts with the double integral in which we consider Riemann sums. Just as one …nds the "area under a curve" for a function y = f (x) of a single variable, we can describe what we mean by area under a surface 3 z = f (x; y) for a function of two variables. It is gratifying that the arithmetic properties such as linearity for an integral of a function of one variable carry over to a double integral, i.e. an integral of a function of two variables. Certain volumes of solids, in which the surfaces involved are plane, are easy to evaluate. s The ideas that are involved in these simple computations lead to Fubini’ theo- rem, Theorem 14.2, which gives the main connection between iterated integrals and double integrals. Riemann sums When we see the need to connect the di¤erent orders of iterated integration over a region, then we see that it becomes important to de…ne the integral: Z f (x; y) dA (7) R Where R is some region in the xy-plane. The easiest, and most obvious way of doing this is to …rst partition the region R into small rectangles P : R1 , R2 , ... Rn (8) We can de…ne inner and outer partitions. The norm, j P j, of the partition is the area of the largest rectangle in the partition. Let A (Ri ) = area of rectangle Ri for i = 1; 2; :::n (9) and choose points in each rectangle: (x1 ; y1 ) in R1 , (x2 ; y2 ) in R2 , ... (xn ; yn ) in Rn (10) De…ne the value of the function at each of these points (these are the "heights" of prisms de…ned by the rectangles, where we allow negative heights to indicate that the prism is below the xy-plane): f1 = f (x1 ; y1 ) , f2 = f (x2 ; x2 ) , ... fn = f (xn ; yn ) (11) Having this setup, we can now de…ne a Riemann sum associated with the integral (7): n X fi A (Ri ) (12) i=1 For numerous types of functions f (x; y), we can de…ne a limiting process for the Riemann sum, to unambiguously specify the value of a double integral. This is notably the case for continuous functions, where the region R is a closed and bounded set in the xy-plane. Then we are able to de…ne Z n X f (x; y) dA = lim fi A (Ri ) (13) R j P j!0 i=1 4 where we are only considering one class of partition, inner partitions or outer partitions. It is then a goal to consider when the Riemann integral will be consistent with either class. Volume Clearly, if f (x; y) > 0, we can interpret the double integral (13) as a volume. It is simple to construct examples using quadratic polynomials, i.e. computing volumes whose bounding surfaces from above are part of a quadric surface. Let y = f (x; y) be a non-negative function on the region R. Then we call Z f dA R the volume of the solid between the region R and the graph of f . Su¢ cient condition for integrability If the integral (13) exists (and is …nite), we say that f is integrable on R. If we restrict our attention to polynomials (or more generally continuous functions), and assume R to be closed and bounded (so that the function values are bounded, and the Riemann sums are …nite), then a su¢ cient condition is not hard to …nd. We merely need to restrict ourselves to unions of a …nite number of nonoverlapping subregions, each of which is vertically or horizontally simple. If R is de…ned by a x b and gL (x) y gU (x), for a …xed value of x, where gL and gU are continuous on [a; b], then the region R is vertically simple. There is an analogous de…nition for a horizontally simple region. Arithmetic properties The arithmetic properties of double integrals are associated with integrations of polynomials, and if we wish to be a bit more general, of rational functions, or continuous functions. For simplicity, assume the following functions are continuous functions, and the region R over which we are integrating is a closed and bounded region in the plane (and composed of nonoverlapping vertically and horizontally simple regions. First, we note that double integrals preserve the vector structure of functions, i.e. they are linear Z Z Z [c1 f1 + c2 f2 ] dA = c1 f1 dA + c2 f2 dA (14) R R R where c1 and c2 are scalars and f1 and f2 are functions. There is also a positivity property: Z jf j dA 0 and equality holds if and only if f = 0 (15) R If we have two nonoverlapping regions R1 and R2 that make the region R, then (inherited from the de…nition of integration in terms of partitions): 5 Z Z Z f dA = f dA + f dA (16) R R1 R2 Due to the positivity property, we have the following ordering property: Z Z if f g then f dA gdA (17) R R Evaluating double integrals If we restrict ourselves to polynomials (more generally, continuous functions), on horizontally or vertically simple regions that are closed and bounded, we s can use Fubini’ theorem to convert a double integral into an iterated integral. s Fubini’ theorem is Theorem 14.2 on page 994. Example 4 on page 996 is an interesting example, because it shows that the choice of order of integration can be important. Example 5 is also worth looking at, because it shows how one can obtain a description for the region R in some situations. Note that this last example applies Wallis’formula from page 536. Integrating with polar coordinates An important consideration involved in integration of a function of one vari- able is the change of variable formula Z Z du f (u) du = f (u (x)) dx (18) dx which is obviously related to the chain rule. We do not consider the general change of variable formula for double integrals in Section 14.3, but instead re- strict ourselves to changing from rectangular to polar coordinates. Integrations over polar coordinates can be split up into integrations over r-simple regions and -simple regions. Riemann sums for polar sectors A polar sector is a region a < r < b and c < <d (19) where for simplicity we require [c; d] to be contained within [0; 2 ]. From this, we can immediately de…ne an r-simple region, where varies between …xed c and d, and for …xed , r varies between the two continuous functions rL ( ) and rU ( ). We can make an analogous de…nition for -simple regions. Then we can consider the integration of continuous functions over nonoverlapping r-simple and -simple regions. The area of a polar sector of the form (19) is approximately r r , where r = b a and = d c, and r is some value selected between a and b, provided that the sector is small (which is a satisfactoryR assumption for Riemann sums). This leads to Theorem 14.3 for expressing R f (x; y) dA in polar coordinates. For example, if R is the polar sector (19), 6 Z Z d Z b f (x; y) dA = f (r cos ; r sin ) rdrd (20) R c a We can obviously write analogous expressions for r-simple of -simple regions. Example 3 on page 1004 is a nice example of the application of polar co- ordinates.to aRproblem that could not be practically integrated using x and y coordinates. R dA is the area of region R. Example 4 provides a nice instance of this applied to polar coordinates. Notice that Z d Z r( ) Z d 1 2 rdrd = (r ( )) d (21) c 0 c 2 Center of mass and moment of inertia Section 14.4 is an application of double integrals to center of mass and mo- ment of inertias of plane laminas. This allows the development of a physical picture for double integration, and in many cases permits us to use notions of symmetry to partially determine quantities such as centers of mass without explicitly carrying out integral expressions. Mass We de…ne mass on the basis of a mass density (x; y), which is a non-negative integrable function, over a closed, bounded region R such as we have discussed before (except now we will refer to it as a plane lamina). The mass m of the lamina R is just the integral of over R. We de…ne the moments of mass Mx and My , as the integrals, respectively, of x and y over R. The quantity Mx My (xcm ; ycm ) = ; (22) m m is called the center of mass or centroid of the lamina. We call Mx and My …rst moments. Moment of inertia We de…ne second moments Ix and Iy to be the integrals over R, respectively, of x2 and y 2 . The sum of Ix and Iy is called the polar moment of inertia and is denoted I0 . Relative to this, we can de…ne a radius of gyration, rgyr : r I rgyr = (23) m where I is the moment of inertia. When I = Ix , we denote rgyr as a y with a double-bar above it, etc. Surface area 7 Section 14.5 concerns another application to the computation of surface area. This is a simple application of the idea that if u and v are vectors along adjacent sides of a parallelogram, then the area of the parallelogram is ju vj. Let y = f (x; y) and consider the points (x; y), (x + dx; y), and (x; y + dy) (points in close proximity). The three points (x; y; f (x; y)) , (x + dx; y; f (x; y) + fx (x; y) dx) , (x; y + dy; f (x; y) + fy (x; y) dy) (24) de…ne a parallelogram that approximates a region of the surface, with vectors (dx; 0; fx (x; y) dx) , (0; dy; fy (x; y) dy) (25) de…ning the sides of the parallelogram. Thus means that a section of surface has the approximate area 2 3 i j k det 4 dx 0 fx dx 5 (26) 0 dy fy dy Therefore, it is not surprising that we de…ne the surface area over the region R to be Z q 2 2 1 + (fx ) + (fy ) dxdy (27) R in cases where it is meaningful to do so. Example 5 on page 1021 is a useful example. It illustrates an application s of Simpson’ rule. Triple integrals Of course, there is no reason, once we understand, to con…ne ourselves to double integrals. It is a simple matter to extend certain elements of our discus- sion to "triple integrals". This is done in Section 14.6. What are the analogies that can readily be pursued? The idea of area computed by double integrals can be extended to volume from triple integrals. The volume under a surface can be extended to a corresponding concept for triple integrals. As this involves four dimensions, the authors avoid discussion of this point. Finally, the center of mass or moments of inertia of a plane lamina can be extended, using triple integrals, to similar concepts for volumes. Change of variables from rectangular to spherical or cylindrical coor- dinates Another analogy is pursued in Section 14.7, where change of coordinates from rectangular to spherical or cylindrical is discussed. This of course corresponds to change of coordinates to polar coordinates for double integrals.Many integrals can be more easily evaluated in cylindrical coordinates or spherical coordinates than in rectangular coordinates. The cylindrical area element is 8 drd dz (28) The spherical area element is sin d d d (29) General change of coordinates In the …nal section, Section 14.8, the problem of general change of coordi- nates is considered. The crucial quantity involved is something that follows how a tiny square in the xy-coordinate system is transformed under a coor- dinate transformation to some coordinates (u; v). Such a tiny square in the xy-coordinate system will essentially become, under many conditions, including polar coordinates (r; ) to a small parallelogram in the uv-plane. It is a determi- nant that measures this shift. The determinant is called the Jacobian, and the whole idea can be extended to any …nite number of variables. Polar, cylindrical and spherical coordinates are by far the most important coordinate frames after the rectangular. However, more general transformations sometimes need to be considered, and this is particularly important for theoretical ideas. Transformation of a rectangle Consider the three vertices of a small parallelogram in uv-space, (u; v), (u + dux ; v + dvx ) and (u + duy ; v + dvy ), corresponding to the rectangle ver- tices (x; y), (x + dx; y) and (x; y + dy) in xy-space, i.e. @u @v dux = dx and dvx = dx (30) @x @x @u @v duy = dy and dvy = dy (31) @y @y with area (@u=@x) dx (@v=@x) dx @u=@x @v=@x det = det dxdy (32) (@u=@y) dy (@v=@y) dy @u=@y @v=@y Jacobian The determinant @ (u; v) @u=@x @v=@x = det (33) @ (x; y) @u=@y @v=@y is called the Jacobian, and we have from the above considerations, the general change of variable formula Z Z @ (u; v) f (u; v) dudv = f (u (x; y) ; v (x; y)) dxdy (34) R(u;v) R(x;y) @ (x; y) 9 where R (x; y) is the region in the xy-plane that corresponds to the region R (u; v) in the uv-plane. Review: Simpson’ rule s Calculators can perform numerical integrations. One practical algorithm s s for this is Simpson’ rule. The basis of Simpson’ rule is Theorem 4.17 on page 311, which is just the integral of a quadratic polynomial between two points expressed in terms of the polynomial: p (x) = Ax2 + Bx + C (35) is our polynomial, and Z b b a a+b p (x) dx = p (a) + 4p + p (b) (36) a 6 2 Here, A, B and C are constants characterizing the polynomial, and [a; b] is just some …nite speci…ed interval. s Simpson’ rule follows by simply treating a function as a concatenation of quadratic polynomials. The result, typically, will be a continuous function that approximates f crudely, and if the intervals [a; b] over which a quadratic ap- proximation is made are fairly small and f is continuous on [a; b], the quadratic concatenation will …t f su¢ ciently well that we will obtain a good approximation to the integral of f over some closed interval. Note that this type of approxi- mation does not necessarily approximate f well: f might be approximated over the small interval by a fairly high degree polynomial. This approximation can be stated explicitly. We refer you to Theorem 4.18 on page 312. When f has continuous fourth derivatives, an error estimate can s be obtained for Simpson’ rule. If we split up the interval of integration (a …nite closed interval) into n parts, then the error is seen to be proportional to 1=n4 and the absolute value of the maximum of the fourth derivative over the interval and the length of the interval. The constant of proportionality is s 1=180, which is not a large number. Thus, Simpson’ rule (without modi…cation to make it computationally more e¢ cient) should converge rather rapidly as n increases (ignoring round-o¤ errors). In terms of computational e¢ ciency, if T is a typical time to compute the function values, then the computational time is on the order of 2nT , which, being proportional to both n and T , is generally not very fast. However, for simple situations, this is not likely to pose a problem. Review: Wallis’formula Wallis’ formula is presented on page 536. It is an expression which gives the value of the integral Z =2 cosn xdx (37) 0 for n 2. The manner in which it is stated by the book presents the formula in a startling way. One of the curiosities of the formula is that when n 10 is even, an extra factor of =2 appears, while when n is odd, it does not. We can present the formula in a more compact way: Z =2 (n 1)!! cosn xdx = E (n) (38) 0 n!! where E (n) takes the value one for n odd, and =2 for n even. The notation n!! signi…es a multiplication: n!! = n (n 2) ::: (39) where we take products of all numbers from one to n skipping every other number and including n. Thus, 5!! = 5 3 1 6!! = 6 4 2 etc. Every such formula like this cannot be regarded as important. However, Wallis’ formula is at least interesting because it is a general pattern. That pattern arises from simple integration by parts. 11