Section 13.1 Double Integrals over Rectangular Regions Page 1

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1 Section. Double Integrals over ectangular egions Page Chapter Preview We have now generalized limits and derivatives to functions of several variables. The next step is to carry out a similar process with respect to integration. As you know, single (one-variable) integrals are developed from iemann sums and are used to compute areas of regions in 2. In an analogous way, we use iemann sums to develop double (two-variable) and triple (three-variable) integrals, which are used to compute volumes of solid regions in. These multiple integrals have many applications in statistics, science, and engineering, including calculating the mass, the center of mass, and moments of inertia of solids with a variable density. Another significant development in this chapter is the appearance of cylindrical and spherical coordinates. These alternative coordinate systems often simplify the evaluation of integrals in three-dimensional space. The chapter closes with the two- and three-dimensional versions of the substitution (change of variables) rule. The overall lesson of the chapter is that we can integrate functions over most geometrical objects, from intervals on the x-axis to regions in the plane bounded by curves to complicated three-dimensional solids.. Double Integrals over ectangular egions In Chapter 2 the concept of differentiation was extended to functions of several variables. In this chapter we extend integration to multivariable functions. By the close of the chapter, we will have completed Table., which is a basic road map for calculus. Table. Derivatives Integrals Single variable: fhxl f 'HxL a b fhxl d x Several variables: fhx, yl and fhx, y, zl f x, f y, f z fhx, yl d A, D fhx, y, zl d V Volumes of Solids Iterated Integrals Average Value Quick Quiz SECTION. EXECISES eview Questions. Write an iterated integral that gives the volume of the solid bounded by the surface fhx, yl= x y over the square =8Hx, yl : 0 x 2, y <. 2. Write an iterated integral that gives the volume of a box with height 0 and base 8Hx, yl : 0 x 5, -2 y 4<.. Write two iterated integrals that equal fhx, yl d A, where =8Hx, yl :-2 x 4, y 5<. CALCULUS: EALY TANSCENDENTALS

2 Section. Double Integrals over ectangular egions Page 2 4. Consider the integral I2 y 2 + x ym d y d x. Give the variable of integration in the first (inner) integral and the - limits of integration. Give the variable of integration in the second (outer) integral and the limits of integration. Basic Skills 5-2. Iterated integrals Evaluate the following iterated integrals x 2 y d x d y H2 x+ yl d x d y 7. 0 pê2 x sin y d y d x 8. 2 Iy 2 + ym d x d y u v d u d v 0. 0 pê2 0 x cos xy d y d x. ln 5 0 ln e x+y d x d y 2. 0 pê4 0 r secqd r dq -9. Iterated integrals Evaluate the following double integrals over the region.. 4. Hx+2 yl d A; =8Hx, yl : 0 x, y 4< Ix 2 + x ym d A; =8Hx, yl : x 2, - y < 5. x y d A; =8Hx, yl : 0 x, y 4< x y sin x 2 d A; =:Hx, yl : 0 x pê2, 0 y > e x+2 y d A; =8Hx, yl : 0 x ln 2, y ln < Ix 4 + y 4 M 2 d A; =8Hx, yl :- x, 0 y < CALCULUS: EALY TANSCENDENTALS

3 Section. Double Integrals over ectangular egions Page 9. Ix 5 - y 5 M 2 d A; =8Hx, yl : 0 x, - y < Choose a convenient order When converted to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral x sec 2 xy d A; =8Hx, yl : 0 x pê, 0 y < x 5 e x y d A; =8Hx, yl : 0 x ln 2, 0 y < y sin xy 2 d A; =:Hx, yl : 0 x, 0 y pê2 > x d A; =8Hx, yl : 0 x 4, y 2< H+ x yl Average value Compute the average value of the following functions over the region. 24. fhx, yl=4- x- y; =8Hx, yl : 0 x 2, 0 y 2< 25. fhx, yl=e -y ; =8Hx, yl : 0 x 6, 0 y ln 2< 26. fhx, yl=sin x sin y; =8Hx, yl : 0 x p, 0 y p< Average value 27. Find the average squared distance between the points of =8Hx, yl :-2 x 2, 0 y 2< and the origin. 28. Find the average squared distance between the points of =8Hx, yl : 0 x, 0 y < and the point H, L. Further Explorations 29. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The region of integration for d x d y is a square. b. If f is continuous on 2, then 4 6 fhx, yl d x d y= 4 6 fhx, yl d y d x. c. If f is continuous on 2, then 4 6 fhx, yl d x d y= 4 6 fhx, yl d y d x. 0. Symmetry Evaluate the following integrals using symmetry arguments. Let =8Hx, yl :-a x a, -b y b<, where a and b are positive real numbers. a. b. x y e -Ix2 +y 2M d A sinhx- yl d A x 2 + y 2 + CALCULUS: EALY TANSCENDENTALS

4 Section. Double Integrals over ectangular egions Page 4. Computing populations The population densities in nine districts of a rectangular county are shown in the figure. a. Use the fact that population =Hpopulation densitylµhareal to estimate the population of the county. b. Explain how the calculation of part (a) is related to iemann sums and double integrals. T 2. Approximating water volume The varying depth of an 8 mµ25 m swimming pool is measured in 5 different rectangles of equal area (see figure). Approximate the volume of water in the pool. -4. Pictures of solids Draw the solid region whose volume is given by the following double integrals. Then find the volume of the solid d y d x I4- x 2 - y 2 M d x d y 5-8. More integration practice Evaluate the following iterated integrals. 5. e 0 x x+ y d y d x x 5 y 2 e x y d y d x y x+ y 2 d x d y CALCULUS: EALY TANSCENDENTALS

5 Section. Double Integrals over ectangular egions Page e y x d y d x Volumes of solids Find the volume of the following solids. 9. The solid between the cylinder fhx, yl=e -x and the region =8Hx, yl : 0 x ln 4, -2 y 2< 40. The solid beneath the plane fhx, yl=6- x-2 y and above the region =8Hx, yl : 0 x 2, 0 y < 4. The solid beneath the plane fhx, yl=24-x-4 y and above the region =8Hx, yl :- x, 0 y 2< CALCULUS: EALY TANSCENDENTALS

6 Section. Double Integrals over ectangular egions Page The solid beneath the paraboloid fhx, yl=2- x 2-2 y 2 and above the region =8Hx, yl : x 2, 0 y < 4. Net volume Let =8Hx, yl : 0 x p, 0 y a<. For what values of a, with 0 a p, is? sinhx+ yl d A equal to Zero average value Let =8Hx, yl : 0 x a, 0 y a<. Find the value of a> 0 such that the average value of the following functions over is zero. 44. fhx, yl= x+ y fhx, yl=4- x 2 - y Maximum integral Consider the plane x+ y+z=6 over the rectangle with vertices at H0, 0L, Ha, 0L, H0, bl, and Ha, bl, where the vertex Ha, bl lies on the line where the plane intersects the xy-plane (so a+ b=6). Find the point Ha, bl for which the volume of the solid between the plane and is a maximum. CALCULUS: EALY TANSCENDENTALS

7 Section. Double Integrals over ectangular egions Page 7 Applications 47. Density and mass Suppose a thin rectangular plate, represented by a region in the xy-plane, has a density given by the function rhx, yl; this function gives the area density in units such as gëcm 2. The mass of the plate is rhx, yl d A. Assume that =8Hx, yl : 0 x pê2, 0 y p< and find the mass of the plates with the following density functions. a. rhx, yl=+sin x b. rhx, yl=+sin y c. rhx, yl=+sin x sin y 48. Approximating volume Propose a method based on iemann sums to approximate the volume of the shed shown in the figure (the peak of the roof is directly above the rear corner of the shed). Carry out the method and provide an estimate of the volume. Additional Exercises 49. Cylinders Let S be the solid in between the cylinder z= fhxl and the region =8Hx, yl : a x b, c y d<, b where fhxl 0 on. Explain why d fhxl d x d y equals the area of the constant cross section of S multiplied by c a Hd - cl, which is the volume of S. 50. Product of integrals Suppose f Hx, yl = ghxl hhyl, where g and h are continuous functions for all real values. a. Show that c d a b fhx, yl d x d y= a b ghxl d x c d hhyl d y. Interpret this result geometrically. b 2 b. Write ghxl d x as an iterated integral. a 0 c. Use the result of part (a) to evaluate 2p Hcos xl e -4 y 2 d y d x An identity Suppose the second partial derivatives of f are continuous on =8Hx, yl : 0 x a, 0 y b<. Simplify 2 f x y d A. 52. Two integrals Let =8Hx, yl : 0 x, 0 y <. CALCULUS: EALY TANSCENDENTALS

8 Section. Double Integrals over ectangular egions Page 8 a. Evaluate cosix y M d A b. Evaluate x y cosix 2 y 2 M d A 5. A generalization Let be as in Exercise 52, let F be an antiderivative of f with FH0L=0 and let G be an antiderivative of F. Show that if f and F are integrable, and r and s are real numbers, then x 2 r- y s- fix r y s M d A= GHL-GH0L. r s CALCULUS: EALY TANSCENDENTALS