{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 291 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 263 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 264 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 265 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 266 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 267 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 268 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 269 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 270 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 271 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 272 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 273 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 5 "C3M11" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 47 "The Evaluation of Triple Integrals by Iterati on" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 167 "The use of triple integral s involves two separate skills, the set up and the evaluation. In or der to set up a triple integral we need to understand a simple phrase, " }{TEXT 256 50 "Surface-to-Surface, Curve-to-Curve, Point-to-Point" }{TEXT -1 329 " . The first or inner integral builds \223columns\224 \+ by adding \223cubes\224 between two surfaces. The second integral add s up those \223columns\224 that are bounded between two curves to obta in \223slabs\224. The third integral adds up the \223slabs\224 betwee n two points to get a volume. This simplistic view of things pre-supp oses that the function " }{XPPEDIT 18 0 "f(x,y,z)=1" "6#/-%\"fG6%%\"x G%\"yG%\"zG\"\"\"" }{TEXT -1 102 " was the integrand. The reader sho uld realize that if the integrand is a density function, such as " } {XPPEDIT 18 0 "delta(x,y,z" "6#-%&deltaG6%%\"xG%\"yG%\"zG" }{TEXT -1 441 " , whose units are \223pounds per cubic foot\224 or \223grams per cubic centimeter\224, then the evaluation of the integral will yield \+ the weight of the solid in pounds or the mass in grams respectively. \+ So triple integrals compute far more than just volume. But the visual ization of computing the volume by adding up small \223cubes\224 as a \+ means of establishing the limits of integration is an essential part o f understanding the process of integration.\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 427 "In the figures that follow from the Maple commands be low (similar to those in C3M8) one sees a vertical column which may be regarded as being formed from solid cubes. If we are finding the vol ume of a solid, then one begins the iterative process by adding up the volume of cubes to obtain the volume of a column. The cubes range be tween two surfaces while two variables are held constant. Here we see change with respect to " }{TEXT 257 3 " z " }{TEXT -1 4 " as " } {TEXT 258 2 " z" }{TEXT -1 19 " ranges between " }{XPPEDIT 18 0 "z= 0" "6#/%\"zG\"\"!" }{TEXT -1 9 " and " }{XPPEDIT 18 0 "z=f(x,y)" " 6#/%\"zG-%\"fG6$%\"xG%\"yG" }{TEXT -1 11 " , while " }{TEXT 260 1 "x " }{TEXT -1 6 " and " }{TEXT 259 1 "y" }{TEXT -1 21 " are held const ant.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "with(plots): with (plottools): c1:=spacecurve([0,y,4-y^2],y=0..2,axes=NORMAL,color=black ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "c2:=spacecurve(\{[x,2 -x,4-x^2-(2-x)^2],[x,x,4-2*x^2]\},x=0..1,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "c3:=spacecurve(\{[x,x,0],[x,2-x,0], [1,1,2*x]\},x=0..1,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "c4:=spacecurve(\{[.38,y,4-.38^2-y^2],[.42,y,4-.42^2-y ^2]\},y=.4..1.6,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "c5:=curve([[.38,.4,3.68],[.42,.4,3.68],[.42,.4,0],[.42,1.6,0],[ .42,1.6,1.28],[.38,1.6,1.28],[.38,1.6,0],[.38,.4,0],[.38,.4,3.68]],col or=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 232 "col:=curve([ [.38,.7,0],[.38,.7,3.32],[.42,.7,3.32],[.42,.7,0],[.38,.7,0],[.38,.74, 0],[.38,.74,3.32],[.42,.74,3.32],[.42,.74,0],[.38,.74,0],[.42,.74,0],[ .42,.7,0],[.42,.7,3.32],[.42,.74,3.32],[.38,.74,3.32],[.38,.7,3.32]],c olor=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "f1:=plot3d([. 38,y,z],y=.7..0.74,z=1..1.04,color=green):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 55 "f2:=plot3d([.42,y,z],y=.7..0.74,z=1..1.04,color=gre en):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "f3:=plot3d([x,.7,z] ,x=.38..0.42,z=1..1.04,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "f4:=plot3d([x,.74,z],x=.38..0.42,z=1..1.04,color=gree n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "f5:=plot3d([x,y,1],x =.38..0.42,y=.7..0.74,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "f6:=plot3d([x,y,1.04],x=.38..0.42,y=.7..0.74,color=gr een):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "display(c1,c2,c3,c 4,c5,col,f1,f2,f3,f4,f5,f6);" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 261 19 "Surface-to-Surface:" }{TEXT -1 25 " green cubes fill column." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "f1:=plot3d([.38,y,z],y=.7..0 .74,z=0..3.32,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "f2:=plot3d([.42,y,z],y=.7..0.74,z=0..3.32,color=green):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "f3:=plot3d([x,.7,z],x=.38..0 .42,z=0..3.32,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "f4:=plot3d([x,.74,z],x=.38..0.42,z=0..3.32,color=green):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "f5:=plot3d([x,y,0],x=.38..0. 42,y=.7..0.74,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "f6:=plot3d([x,y,3.32],x=.38..0.42,y=.7..0.74,color=green):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "display(c1,c2,c3,c4,c5,f1,f2 ,f3,f4,f5,f6);" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 262 15 "Curve-to-Cur ve:" }{TEXT -1 25 " green columns fill slab." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "f1:=plot3d([.38,y,z],y=.4..1.6,z=0..4-.4^2-y^2,c olor=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "f2:=plot3d( [.42,y,z],y=.4..1.6,z=0..4-.4^2-y^2,color=green):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 55 "f3:=plot3d([x,.4,z],x=.38..0.42,z=0..3.68,co lor=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "f4:=plot3d([ x,1.6,z],x=.38..0.42,z=0..1.28,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "f5:=plot3d([x,y,0],x=.38..0.42,y=.4..1.6,color=g reen):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "f6:=plot3d([x,y,4 -.4^2-y^2],x=.38..0.42,y=.4..1.6,color=green):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 44 "c6:=curve([[1,.5,0],[1,1.5,0]],color=BLACK):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "display(c1,c2,c3,c4,c5,c6 ,f1,f2,f3,f4,f5,f6);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 263 16 "Point- to-Point: " }{TEXT -1 24 " green slabs fill solid." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 595 "There is a very important idea that we need to u nderstand from the beginning. If the integrand is continuous over the region or solid, then the integrand has absolutely nothing to do with establishing the limits of integration of the triple integral. These limits are determined solely by the geometry of the solid involved. \+ It is the relationship of the solid to the coordinate planes that eith er permits the integral to be set up as one integral or it forces two \+ or more integrals to be used. There are three variables, so there are six orders of integration that are possible, including " }{TEXT 264 10 "dx dy dz " }{TEXT -1 5 "and " }{TEXT 265 8 "dy dz dx" }{TEXT -1 510 " and so forth. Having said that, the process of successive anti -differentiation may range from easy to impossible when different poss ible orders are considered. We begin with a picture of a solid figure . After the first integral we look at that solid so that our line of \+ sight is parallel to the columns we just built. We now see a two-dime nsional view of the solid with the ends of the columns appearing as sq uares. We have reduced a triple integral to a double integral, which \+ we know how to evaluate.\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 572 "In a previous chapter we learned how to find partial derivatives. Essen tially this involved differentiating with respect to one variable whil e the other variables were held constant. To evaluate a triple integr al in iterated form we anti-differentiate with respect to one variable while the others are held constant, evaluate at the (variable) endpoi nts, and repeat this process with respect to a different variable. Fo r all practical purposes we are \223partially integrating\224 instead \+ of \223partially differentiating\224. If we begin by anti-differenti ating with respect to " }{TEXT 266 1 "z" }{TEXT -1 66 " , then the int egration takes place between two functions, say " }{XPPEDIT 18 0 "z =g(x,y)" "6#/%\"zG-%\"gG6$%\"xG%\"yG" }{TEXT -1 8 " and " } {XPPEDIT 18 0 "z=h(x,y)" "6#/%\"zG-%\"hG6$%\"xG%\"yG" }{TEXT -1 22 " . It would look like" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 4 " " } {XPPEDIT 18 0 "Int(Int(F(x,y,h(x,y))-F(x,y,g(x,y)),y=(y=r(x))..(y=s(x) )),x=(x=a)..(x=b))" "6#-%$IntG6$-F$6$,&-%\"FG6%%\"xG%\"yG-%\"hG6$F,F- \"\"\"-F*6%F,F--%\"gG6$F,F-!\"\"/F-;/F--%\"rG6#F,/F--%\"sG6#F,/F,;/F,% \"aG/F,%\"bG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "after the first a nti-differentiation. The curves " }{XPPEDIT 18 0 "y=r(x)" "6#/%\"yG -%\"rG6#%\"xG" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "y=s(x)" "6#/%\"y G-%\"sG6#%\"xG" }{TEXT -1 51 " were seen in outline form as we looke d down the " }{TEXT 267 1 "z" }{TEXT -1 40 "-axis after integrating wi th respect to " }{TEXT 268 1 "z" }{TEXT -1 61 " . It is important to \+ note that after the substitution of " }{XPPEDIT 18 0 "z=g(x,y)" "6#/ %\"zG-%\"gG6$%\"xG%\"yG" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "z=h(x, y)" "6#/%\"zG-%\"hG6$%\"xG%\"yG" }{TEXT -1 8 " into " }{XPPEDIT 18 0 "F(x,y,z)" "6#-%\"FG6%%\"xG%\"yG%\"zG" }{TEXT -1 19 " , the variabl e " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 48 " disappears and a do uble integral in terms of " }{TEXT 269 1 "x" }{TEXT -1 7 " and " } {TEXT 270 1 "y" }{TEXT -1 11 " remains.\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 271 9 "\001Example " }{TEXT -1 35 "1 Suppose a solid is bounde d by " }{XPPEDIT 18 0 "y+z=4" "6#/,&%\"yG\"\"\"%\"zGF&\"\"%" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "z=0" "6#/%\"zG\"\"!" }{TEXT -1 3 " , " } {XPPEDIT 18 0 "y=x^2" "6#/%\"yG*$%\"xG\"\"#" }{TEXT -1 3 " , " } {XPPEDIT 18 0 "y=3" "6#/%\"yG\"\"$" }{TEXT -1 112 " . Find the volume of the solid by using an iterated integral. The first or inner integ ral will have a roof " }{XPPEDIT 18 0 "y+z=4" "6#/,&%\"yG\"\"\"%\"zG F&\"\"%" }{TEXT -1 16 " and a floor " }{XPPEDIT 18 0 "z=0" "6#/%\"z G\"\"!" }{TEXT -1 88 " as bounds. Because \223floor \224is a protecte d word in Maple, we will use \223floor1\224 instead." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(student): with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "roof:=plot3d([x,y,4-y],x=-sqrt(3).. sqrt(3),y=x^2..3,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "floor1:=plot3d([x,y,0],x=-sqrt(3)..sqrt(3),y=x^2..3,color=green) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "display(roof,floor1); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "The integral for this part is " }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(1,z= (z=0)..(z=4-y))=4-y" "6#/-%$IntG6$\"\"\"/%\"zG;/F)\"\"!/F),&\"\"%F'%\" yG!\"\",&F/F'F0F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Now we add t he curved side " }{XPPEDIT 18 0 "y=x^2" "6#/%\"yG*$%\"xG\"\"#" } {TEXT -1 27 " and the vertical plane " }{XPPEDIT 18 0 "y=3" "6#/%\" yG\"\"$" }{TEXT -1 42 " as well as two lines which show where " } {XPPEDIT 18 0 "x=-sqrt(3)" "6#/%\"xG,$-%%sqrtG6#\"\"$!\"\"" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "x=sqrt(3)" "6#/%\"xG-%%sqrtG6#\"\"$" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "cside:=plo t3d([x,x^2,z],x=-sqrt(3)..sqrt(3),z=0..4-x^2,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "vside:=plot3d([x,3,z],x=-sqrt(3)..s qrt(3),z=0..1,color=magenta):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "L1:=spacecurve([-sqrt(3),y,0],y=-.5..4,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "L2:=spacecurve([sqrt(3),y,0 ],y=-.5 ..4,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "displa y(roof,floor1,cside,vside,L1,L2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Now we integrate between the curves " }{XPPEDIT 18 0 "y=x^2" "6#/ %\"yG*$%\"xG\"\"#" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "y=3" "6#/%\"y G\"\"$" }{TEXT -1 1 "." }}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "Int(4-y,y)=4*y-y^2/2" "6#/-%$IntG6$,&\"\"%\"\"\"%\"yG! \"\"F*,&*&F(F)F*F)F)*&F*\"\"#F/F+F+" }{TEXT -1 10 " so " } {XPPEDIT 18 0 "Int(4-y,y=(y=x^2)..(y=3))" "6#-%$IntG6$,&\"\"%\"\"\"%\" yG!\"\"/F);/F)*$%\"xG\"\"#/F)\"\"$" }{TEXT -1 6 " = (" }{XPPEDIT 18 0 "(12-9/2)" "6#,&\"#7\"\"\"*&\"\"*F%\"\"#!\"\"F)" }{TEXT -1 6 " ) - ( " }{XPPEDIT 18 0 "4x^2-x^4/2" "6#,&*&\"\"%\"\"\"*$%\"xG\"\"#F&F&*&F(F% F)!\"\"F+" }{TEXT -1 7 " ) = " }{XPPEDIT 18 0 "15/2-4*x^2+x^4/2" "6# ,(*&\"#:\"\"\"\"\"#!\"\"F&*&\"\"%F&*$%\"xGF'F&F(*&F,F*F'F(F&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "It remains to integrate this last \+ result between the " }{TEXT 272 1 "x" }{TEXT -1 13 " values of \261 " }{XPPEDIT 18 0 "sqrt(3)" "6#-%%sqrtG6#\"\"$" }{TEXT -1 2 " ." }}} {EXCHG {PARA 264 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(15/2-4*x ^2+x^4/2,x=-sqrt(3)..sqrt(3))=44/5*sqrt(3)" "6#/-%$IntG6$,(*&\"#:\"\" \"\"\"#!\"\"F**&\"\"%F**$%\"xGF+F*F,*&F0F.F+F,F*/F0;,$-%%sqrtG6#\"\"$F ,-F66#F8*(\"#WF*\"\"&F,-F66#F8F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "The triple integral that accomplishes the same thing is" }}} {EXCHG {PARA 265 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(Int(Int( 1,z=(z=0)..(z=4-y)),y=(y=x^2)..(y=3)),x=(x=-sqrt(3))..(x=sqrt(3))" "6# -%$IntG6$-F$6$-F$6$\"\"\"/%\"zG;/F,\"\"!/F,,&\"\"%F*%\"yG!\"\"/F3;/F3* $%\"xG\"\"#/F3\"\"$/F9;/F9,$-%%sqrtG6#F " 0 "" {MPLTEXT 1 0 54 "Q:=Tripleint(1,z=0..4-y,y=x^2..3,x=-sqrt(3) ..sqrt(3));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(Q);" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "Compare the Maple syntax for the triple integral with that for drawing \221roof \222and \221floor1\222 . The order of the limits in \221" }{TEXT 273 9 "Tripleint" }{TEXT -1 96 "\222 determines the order of the variables to be integrated. N ow let\222s consider an example where " }{TEXT 274 1 "z" }{TEXT -1 121 " is not the first variable to be integrated. In fact, let\222s \+ really analyze how we must select the order of integration." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 275 10 "Exampl e 2 " }{TEXT -1 68 " Set up an iterated triple integral to find the tr iple integral of " }{XPPEDIT 18 0 "f(x,y,z)=x" "6#/-%\"fG6%%\"xG%\"yG %\"zGF'" }{TEXT -1 18 " over the solid " }{TEXT 276 1 "S" }{TEXT -1 23 " which is bounded by " }{XPPEDIT 18 0 "z=2*x^2" "6#/%\"zG*&\"\"# \"\"\"*$%\"xGF&F'" }{TEXT -1 5 " , " }{XPPEDIT 18 0 "x+y+z=4" "6#/,( %\"xG\"\"\"%\"yGF&%\"zGF&\"\"%" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "x+z =3" "6#/,&%\"xG\"\"\"%\"zGF&\"\"$" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "x =0" "6#/%\"xG\"\"!" }{TEXT -1 8 " , and " }{XPPEDIT 18 0 "y=0" "6#/% \"yG\"\"!" }{TEXT -1 4 " . " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 166 " Executing the following commands give a view of the solid to show the \+ three surfaces which do not lie in a coordinate plane and the three vi ews down each of the axes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(student): with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "bottom:=plot3d([x,y,2*x^2],x=0..1,y=0..4-x-2*x^2,color=blue): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "top:=plot3d([x,y,3-x],x =0..1,y=0..1,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "slant:=plot3d([x,4-x-z,z],x=0..1,z=2*x^2..3-x,color=cyan):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "left:=plot3d([x,0,z],x=0..1, z=2*x^2..3-x,color=magenta):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "back:=plot3d([0,y,z],z=0..3,y=0..4-z,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "display(bottom,top,slant,left,back, axes=NORMAL);" }}}{EXCHG {PARA 266 "" 0 "" {TEXT -1 13 " the solid " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "display(bottom,top,slan t,left,back,orientation=[0,90]);" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 10 " down the " }{TEXT 277 1 "x" }{TEXT -1 5 "-axis" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "From our view of the solid, we realize th at we may not just assume that " }{TEXT 280 1 "z" }{TEXT -1 31 " is \+ first. Looking down the " }{TEXT 281 1 "x" }{TEXT -1 128 "-axis we s ee three different surfaces. A light beam from behind the figure woul d always enter through the single surface where " }{XPPEDIT 18 0 "x=0 " "6#/%\"xG\"\"!" }{TEXT -1 136 " . But as it exits towards us, it co uld leave through three different surfaces, which denies the choice of integrating with respect to " }{TEXT 282 1 "x" }{TEXT -1 31 " first. \+ Then we look down the " }{TEXT 283 1 "z" }{TEXT -1 28 "-axis from abo ve the figure." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "display(b ottom,top,slant,left,back,orientation=[0,0]);" }}}{EXCHG {PARA 268 "" 0 "" {TEXT -1 8 "down the" }{TEXT 278 2 " z" }{TEXT -1 5 "-axis" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "display(bottom,top,slant,lef t,back,orientation=[90,90]);" }}}{EXCHG {PARA 269 "" 0 "" {TEXT -1 8 " down the" }{TEXT 279 2 " y" }{TEXT -1 5 "-axis" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "We see that a light beam entering the solid from be low the figure would always enter through the parabolic surface " } {XPPEDIT 18 0 "z=2*x^2" "6#/%\"zG*&\"\"#\"\"\"*$%\"xGF&F'" }{TEXT -1 53 ". That beam may exit through either of two planes, " }{XPPEDIT 18 0 "x+z=3" "6#/,&%\"xG\"\"\"%\"zGF&\"\"$" }{TEXT -1 6 " or " } {XPPEDIT 18 0 "x+y+z=4" "6#/,(%\"xG\"\"\"%\"yGF&%\"zGF&\"\"%" }{TEXT -1 61 " . Because there are two exit surfaces we should not choose " }{TEXT 284 2 " z" }{TEXT -1 167 " as the first variable of integratio n - unless we want to break the integral up into two distinct integral s over solids which abut. Last, and best, we look down the " }{TEXT 285 1 "y" }{TEXT -1 69 "-axis and realize that a light from behind wou ld enter in the plane " }{XPPEDIT 18 0 "y=0" "6#/%\"yG\"\"!" }{TEXT -1 42 " and exit towards us through the plane " }{XPPEDIT 18 0 "x+y +z=4" "6#/,(%\"xG\"\"\"%\"yGF&%\"zGF&\"\"%" }{TEXT -1 79 " . Ahah! Ou r first choice should (must) be to integrate first with respect to " } {TEXT 286 2 " y" }{TEXT -1 112 " and build columns from left to right (when looking at our first view of the solid) which are parallel to t he " }{TEXT 287 1 "y" }{TEXT -1 85 "-axis. We begin to build our int egral and show the variables in the limits at first:" }}}{EXCHG {PARA 270 "" 0 "" {TEXT -1 40 "First step: The innermost integral is " } {XPPEDIT 18 0 "Int(x,y=(y=0)..(y=4-x-y)" "6#-%$IntG6$%\"xG/%\"yG;/F(\" \"!/F(,(\"\"%\"\"\"F&!\"\"F(F0" }{TEXT -1 2 " " }}{PARA 271 "" 0 "" {TEXT -1 54 "It's followed by a double integral with differential " } {TEXT 288 8 "dA=dx dz" }{TEXT -1 6 " or " }{TEXT 289 8 "dA=dz dx" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Now, and this is important, retur n to the view down the " }{TEXT 290 1 "y" }{TEXT -1 487 "-axis produce d above and look at this as if it were two-dimensional and is the regi on for a double integral. In our first integral we went \221surface-t o-surface\222, and now we must go \221curve-to-curve\222 as before. I ntegrating right to left (positive direction) would not work because t he light beam would exit through two different curves. Our only choic e is vertical, and the light beam test allows the light to enter throu gh the parabola and exit through the line when the light is below." }} }{EXCHG {PARA 272 "" 0 "" {TEXT -1 46 "Second step: The first two in tegrals are " }{XPPEDIT 18 0 "Int(Int(x,y=(y=0)..(y=4-x-y)),z=(z=2* x^2)..(z=3-x))" "6#-%$IntG6$-F$6$%\"xG/%\"yG;/F*\"\"!/F*,(\"\"%\"\"\"F (!\"\"F*F2/%\"zG;/F4*&\"\"#F1*$F(F8F1/F4,&\"\"$F1F(F2" }{TEXT -1 42 " \+ and the full triply iterated integral is" }}}{EXCHG {PARA 273 "" 0 " " {TEXT -1 13 "Third step: " }{XPPEDIT 18 0 "Int(Int(Int(x,y=(y=0)..( y=4-x-y)),z=(z=2*x^2)..(z=3-x)),x=(x=0)..(x=1))" "6#-%$IntG6$-F$6$-F$6 $%\"xG/%\"yG;/F,\"\"!/F,,(\"\"%\"\"\"F*!\"\"F,F4/%\"zG;/F6*&\"\"#F3*$F *F:F3/F6,&\"\"$F3F*F4/F*;/F*F//F*F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Now consider the Maple syntax for this triple integral." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "A2:=Tripleint(x,y=0..4-x-z,z =2*x^2..3-x,x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "val ue(A2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 291 14 "C3M11 Problems" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 188 "Evaluate the triple integrals in \+ 1, 2, and 3 by pencil and paper and by Maple to check your answers. U se Maple to plot the solid figure which is the domain of the integral. Hint: Insert " }{TEXT 292 1 "x" }{TEXT -1 4 " =, " }{TEXT 293 1 "y" }{TEXT -1 4 " =, " }{TEXT 294 2 "z " }{TEXT -1 165 "= appropriately i n the limits of the integrals before you begin, to help find equations for the surfaces and curves. Ignore the integrand when sketching the solid." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 " 1. " }{XPPEDIT 18 0 "A=Int(Int(Int(y*z*cos(x*y),z=1..4),x=0..Pi),y=-1..2)" "6#/%\"AG-%$Int G6$-F&6$-F&6$*(%\"yG\"\"\"%\"zGF.-%$cosG6#*&%\"xGF.F-F.F./F/;F.\"\"%/F 4;\"\"!%#PiG/F-;,$F.!\"\"\"\"#" }{TEXT -1 26 " 2. " }{XPPEDIT 18 0 "B=Int(Int(Int(x,z=0..6-x-y),y=0..3-x),x=0..2)" "6# /%\"BG-%$IntG6$-F&6$-F&6$%\"xG/%\"zG;\"\"!,(\"\"'\"\"\"F,!\"\"%\"yGF4/ F5;F0,&\"\"$F3F,F4/F,;F0\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 " \+ 3. " }{XPPEDIT 18 0 "C=Int(Int(Int(x,z=0..4-2*y),y=0..sqrt(4-x^2)),x =0..2)" "6#/%\"CG-%$IntG6$-F&6$-F&6$%\"xG/%\"zG;\"\"!,&\"\"%\"\"\"*&\" \"#F3%\"yGF3!\"\"/F6;F0-%%sqrtG6#,&F2F3*$F,F5F7/F,;F0F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "In the remaining problems, use Maple to \+ sketch the solid and to find the volume by evaluating the triple integ ral ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 " 4. The solid is bound ed by " }{XPPEDIT 18 0 "x-y+z=2" "6#/,(%\"xG\"\"\"%\"yG!\"\"%\"zGF&\" \"#" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "x=z^2" "6#/%\"xG*$%\"zG\"\"#" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "x+z=2" "6#/,&%\"xG\"\"\"%\"zGF&\"\"# " }{TEXT -1 3 " , " }{XPPEDIT 18 0 "y=0" "6#/%\"yG\"\"!" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 " 5. The solid is bounded by " } {XPPEDIT 18 0 "z=2-x^2" "6#/%\"zG,&\"\"#\"\"\"*$%\"xGF&!\"\"" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "x=z" "6#/%\"xG%\"zG" }{TEXT -1 3 " , " } {XPPEDIT 18 0 "x+y+z=3" "6#/,(%\"xG\"\"\"%\"yGF&%\"zGF&\"\"$" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 3 " , " } {XPPEDIT 18 0 "y=0" "6#/%\"yG\"\"!" }{TEXT -1 5 " . " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 " 6. The solid is bounded by " } {XPPEDIT 18 0 "z-y=2" "6#/,&%\"zG\"\"\"%\"yG!\"\"\"\"#" }{TEXT -1 3 " \+ , " }{XPPEDIT 18 0 "y=2" "6#/%\"yG\"\"#" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "y+z=2" "6#/,&%\"yG\"\"\"%\"zGF&\"\"#" }{TEXT -1 3 " , " } {XPPEDIT 18 0 "x-y^2=2" "6#/,&%\"xG\"\"\"*$%\"yG\"\"#!\"\"F)" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 5 " . " }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 " 7. Let k be an instructor \+ assigned constant as described in the preface. The solid is bounded b y " }{XPPEDIT 18 0 "y = x^2;" "6#/%\"yG*$)%\"xG\"\"#\"\"\"" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "x = y^2;" "6#/%\"xG*$)%\"yG\"\"#\"\"\"" } {TEXT -1 3 " , " }{XPPEDIT 18 0 "z = 0;" "6#/%\"zG\"\"!" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "z = kx;" "6#/%\"zG%#kxG" }{TEXT -1 23 " . \+ " }}}}{MARK "96" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }