Triple Integrals \302\251 2006 Mike May, S. J.- maymk@slu.edu restart:with(Student[MultivariateCalculus]):with(plots): There are several related skills we need for doing triple integrals 1) Given a triple integral, be able to evaluate the integral; 2) Given a region described by surfaces, convert to a region of integration; 3) Given a triple integral, be able to visualize the region of integration; (Task 3 is the reverse of task 2.) 4) Given a triple integral, be able to change the order of integration; 5) Explain the triple integral as adding a function over points arranged in lines arranged in sections of planes arranged in a region of integration.

The problems of triple integrals are very much like the problems of double integrals, only with three steps rather than two. The first problem is to set up the limits of integration. When we did double integrals, the limits of the inside variable were functions of the outside variable. A region could be set up in 2 ways with the 2 possible orders to the variables. With triple integrals we now have a variable inside two other variables, with limits that are functions of those two variables. There are now 6 possible ways to order the variables. If we set up the integral to integrate f(x,y,z) with respect to z first, then with respect to y, then with respect to x, the general form will be 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, where, over a particular (x,y), z is restricted between g(x,y) and h(x,y) and, for a particular x, y is restricted between s(x) and t(x). We can use Maple to check triple integrals, using the int command three times. If we want to integrate the function f(x,y,z) = x + y^2 + z^3 over the cube bounded by the planes x=1, x=3, y=-1, y=2, z=0, and z=1, we evaluate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2KS1JKG1zdWJzdXBHRiQ2Jy1JI21vR0YkNi1RJiZpbnQ7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSIwRidGOC1GUDYkUSIxRidGOC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIyRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YrLUYjNiktRjI2J0Y0LUYjNiQtRjU2LVEqJnVtaW51czA7RidGOEY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORmBvRlMtRlA2JFEiMkYnRjhGVkZZRistRiM2KC1GMjYnRjRGUy1GUDYkUSIzRidGOEZWRlktSShtZmVuY2VkR0YkNiQtRiM2KC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjlRJ2l0YWxpY0YnLUY1Ni1RIitGJ0Y4RjtGPkZARkJGREZGRkhGX29GYW8tRiM2Iy1JJW1zdXBHRiQ2JS1GLDYlUSJ5RidGZHBGZ3BGYm8vRldGZW5GaXAtRiM2Iy1GX3E2JS1GLDYlUSJ6RidGZHBGZ3BGaW9GZHFGK0Y4RistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuM2VtRicvJSZkZXB0aEdGYXIvJSpsaW5lYnJlYWtHUSVhdXRvRictRjU2LVEwJkRpZmZlcmVudGlhbEQ7RidGOEY7Rj5GQEZCRkRGRkZIRkpGTUZhcEYrRlxyRmpyRmFxRitGXHJGanJGaXFGKw== by simply executing the command: Int(Int(Int(x+y^2+z^3, x=1..3), y=-1..2),z=0..1)= int(int(int(x+y^2+z^3, x=1..3), y=-1..2),z=0..1); Note that we use the Int command (upper case I) to be able to check the integral that is being evaluated. (It is also useful to note the command used "=" to give an equation, showing expressions on both sides, rather than ":=" to make an assignment to the name on the left side.) At times it is useful to see the intermediate steps. Int(Int(Int(x+y^2+z^3, x=1..3), y=-1..2),z=0..1)= Int(Int(int(x+y^2+z^3, x=1..3), y=-1..2),z=0..1); Int(Int(int(x+y^2+z^3, x=1..3), y=-1..2),z=0..1)= Int(int(int(x+y^2+z^3, x=1..3), y=-1..2),z=0..1); Int(int(int(x+y^2+z^3, x=1..3), y=-1..2),z=0..1)= int(int(int(x+y^2+z^3, x=1..3), y=-1..2),z=0..1); We can do a similar check with the MultiInt command from the Student[MultivariateCalculus] package. answer = MultiInt(x+y^2+z^3,x=1..3,y=-1..2, z=0..1, output = value); setup = MultiInt(x+y^2+z^3,x=1..3,y=-1..2, z=0..1, output = integral); print("A worked solution"); MultiInt(x+y^2+z^3,x=1..3,y=-1..2, z=0..1, output = steps); If we have not invoked the Student[MultivariateCalculus] package using the with command at the start of the worksheet, we can still use the command in the long form. answer = Student[MultivariateCalculus][MultiInt](x+y^2+z^3,x=1..3,y=-1..2, z=0..1,output = steps);

1) Evaluate the triple integrals of the following functions over the indicated regions. Follow the model used above so that you show both the unevaluated integral and the evaluated expression together. (a) f(x, y, z) = x^2 +5*y^2 - z. W is the rectangular box 0 \342\211\244 x \342\211\244 2, -1 \342\211\244 y \342\211\244 1, 2 \342\211\244 z \342\211\244 3. (b) h(x, y, z) = a*x + b*y + c*z. W is the rectangular box 0 \342\211\244 x \342\211\244 1, 0 \342\211\244 y \342\211\244 1, 0 \342\211\244 z \342\211\244 2. (c) f(x, y, z) = exp(-x-y-z). W is the rectangular box 0 \342\211\244 x \342\211\244 a, 0 \342\211\244 y \342\211\244 b, 0 \342\211\244 z \342\211\244 c.

When doing triple integral problems the first task is to often to identify the region of integration when it is defined by a number of boundary surfaces, and transform those surfaces into limits of integration. Consider the surface bounded by the curves 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjNGJ0Y5Rjk=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjNGJ0Y5LUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GVy1GLDYlUSJ5RidGL0YyRjk=, and 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. The easiest way to get a first view of the region is with the implicitplot3d command, which is part of the plots package. To help us understand what we are looking at we color the surfaces. eqn1 := x^2+y^2=16: eqn2 := y=3: eqn3 := z=3+y: eqn4 := z=-3-2*y: implicitplot3d([eqn1, eqn2, eqn3, eqn4], x=-5..5, y=-5..5, z=-10..10, axes=boxed, color=[red, yellow, pink, blue]); First we want to find an axis so that looking down that axis we have a single front and a single back to the region. The axis with this property will be the first variable of integration. In this case the top is pink (eqn3) and the bottom is blue (eqn4). For every (x0,y0) in the region we will integrate from z=-3-2*y to z=3+y Next we want to look down the z axis (or which ever variable we are integrating first) and see the boundary of the shadow of the region in the xy-plane. Algebraically we look at where the front and back surfaces intersect other surfaces by solving for z and substituting into other equations. In this case equations 1 and 2 are unchanged. The top and bottom intersect along the line y=-2. solve({z=-3-2*y, z=3+y}); Now we plot the three curves with implicitplot to get the region of integration for the shadow. eqn1 := x^2+y^2=16: eqn2 := y=3: eqn5 := -3-2*y=3+y: implicitplot([eqn1, eqn2, eqn5], x=-5..5, y=-5..5, axes=boxed, color=[red, yellow, blue]); Looking at this region of integration we next want to integrate with respect to x so that the region has a single left curve and a single right curve. Thus we need to solve eqn1 for x and make the two choices for the two sides of the region. We would integrate x from 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 to 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 Looking at the shadow of the region on the y axis we then integrate y from y=-2 to y=3. Thus we are ready to set up the triple integral. Int(Int(Int(f,z=-3-2*y..3+y), x=-sqrt(16-y^2)..sqrt(16-y^2)), y=-2..3);

2) Find the limits of integration for the regions described below: (a) The region bounded by the planes, x=0, y=0, z=0, x+y=1, and y+2z=2. (b) The wedge cut from the cylinder 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 by the planes z=0 and z=-y, with y\342\211\2440. (c) The region bounded by x=0, y=0, z=0, 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, and 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.

In setting up the limits of integration above we gave the order of integration and the 6 limits of integration. intorder := [z,x,y]; firstlo:= -3-2*y: firsthi:= 3+y: secondlo := -sqrt(16-y^2): secondhi := sqrt(16-y^2): thirdlo:= -2: thirdhi:= 3: Notice that the limits for each integration are functions in the remaining variables. That means the inside limits can use the outside variables while the outside limits are constants. We then use the Int command to make sure we are asking for the integral we think we are integrating for. Int(Int(Int(f,intorder[1]=firstlo..firsthi), intorder[2]=secondlo..secondhi), intorder[3]=thirdlo..thirdhi); Next we use some technical commands that let us parameterize the surfaces of the region of integration. (Parameterized surfaces is a topic for later in the course, so you are not expected to do this on your own. We parameterize from the outside in. We use s with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJiZsZXE7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0YyLUYsNiRRIjFGJ0YvRi8= to define a value of third0 that is between the high and low values of the third variable. For a specified value of the third variable we use t with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJiZsZXE7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0YyLUYsNiRRIjFGJ0YvRi8= to specify a value of the second variable. With the outer variables specified we use u with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJiZsZXE7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0YyLUYsNiRRIjFGJ0YvRi8= to specify a value of the first variable between the front and back.) third0 := s -> s*thirdhi+(1-s)*thirdlo: second0 := (s,t) -> t*eval(secondhi,intorder[3]=third0(s))+ (1-t)*eval(secondlo,intorder[3]=third0(s)): firstfront := (s,t) -> eval(firsthi, {intorder[2]=second0(s,t), intorder[3]=third0(s)}): firstback := (s,t) -> eval(firstlo, {intorder[2]=second0(s,t), intorder[3]=third0(s)}): first0 := (s,t,u) -> u*firstfront(s,t)+ (1-u)*firstback(s,t): We also want a technical function for labels of our axes based on integration order. maplabels := x -> [convert(x[3],string),convert(x[2],string), convert(x[1],string)]: maplabels(intorder); Now we are ready to plot the region of integration. We need six parameterized surfaces for the top and bottom, left and right, and front and back of the region. frontback := plot3d([ [third0(s), second0(s,t),first0(s,t,1)], [third0(s), second0(s,t),first0(s,t,0)]], s=0..1, t=0..1,color=[red,yellow]): leftright := plot3d([ [third0(s), second0(s,1),first0(s,1,u)], [third0(s), second0(s,0),first0(s,0,u)]], s=0..1, u=0..1,color=[green, blue]): topbottom := plot3d([ [third0(1), second0(1,t),first0(1,t,u)], [third0(0), second0(0,t),first0(0,t,u)]], t=0..1, u=0..1,color=[pink, orange]): display3d({frontback, topbottom, leftright}, axes=boxed, style=patchcontour, scaling=constrained, labels=maplabels(intorder)); To look cleanly down one axis, front and back, we check orientations [0,0] and [0,180] in the first direction, [0,90] and [180,90] in the second direction, and [90,90] and [-90,90] in the third direction. If we have a "good" first direction, we only see front in one direction and only see back in the other direction. We also don't cut out anything between the front and back. Rotating the region, we can see that although there are six limits of integration, there are only four surfaces. Rotating the graph to look down each axis in turn, we have a single front and back surface in the z direction (our original order) and in the x direction. Looking down the y axis we see several colors so y is not a good first variable of integration.

We want to create two functions so we can check visualizations. IntCheck := (intorder, firstlo, firsthi, secondlo, secondhi, thirdlo, thirdhi) -> Int(Int(Int(f,intorder[1]=firstlo..firsthi), intorder[2]=secondlo..secondhi), intorder[3]=thirdlo..thirdhi): IntVisual := proc(intorder, firsthi, firstlo, secondhi, secondlo, thirdhi,thirdlo, transpar) local third0, second0, firstfront, firstback, first0, maplabels, topbottom, leftright, frontback: third0 := s -> s*thirdhi+(1-s)*thirdlo; second0 := (s,t) -> t*eval(secondhi,intorder[3]=third0(s))+ (1-t)*eval(secondlo,intorder[3]=third0(s)); firstfront := (s,t) -> eval(firsthi, {intorder[2]=second0(s,t), intorder[3]=third0(s)}): firstback := (s,t) -> eval(firstlo, {intorder[2]=second0(s,t), intorder[3]=third0(s)}): first0 := (s,t,u) -> u*firstfront(s,t)+ (1-u)*firstback(s,t): maplabels := x -> [convert(x[3],string),convert(x[2],string), convert(x[1],string)]: maplabels(intorder); frontback := plot3d([ [third0(s), second0(s,t),first0(s,t,1)], [third0(s), second0(s,t),first0(s,t,0)]], s=0..1, t=0..1,color=[red,yellow]): leftright := plot3d([ [third0(s), second0(s,1),first0(s,1,u)], [third0(s), second0(s,0),first0(s,0,u)]], s=0..1, u=0..1,color=[green, blue]): topbottom := plot3d([ [third0(1), second0(1,t),first0(1,t,u)], [third0(0), second0(0,t),first0(0,t,u)]], t=0..1, u=0..1,color=[pink, orange]): display3d({frontback, topbottom, leftright}, axes=boxed, style=patchnogrid, labels=maplabels(intorder), transparency=transpar); end proc: We are interested in a region bounded by the surfaces, z=x^2+y^2+3, x+y-z=0, x+y=-2, x-y=2, y=5 and y=-1. We need to do some playing with the viewing window to get the shape. implicitplot3d([z=x^2+y^2+3, x+y-z=0,x+y=-2,x-y=2,y=5, y=-1], x=-8..8, y=-5..6, z=-5..80, axes=normal, transparency=0, color=[red, green, blue, yellow, brown, pink], style=patchnogrid); Once again it is easiest to do z first since there is a simple top and bottom. Looking down the z axis, we then have a truncated triangle and it is easiest to integrate in x next since we then have a single line for left and right. Finally we integrate in y. We check to see that we have set up the integral correctly. Now we can simply define the problem, then check the integral, and check the visualization. intorder := [z,x,y]: firsthi:= 3+x^2+2*y^2: firstlo:= x+y: secondhi := 2+y: secondlo := -2-y: thirdhi:= 5: thirdlo:= -1: IntCheck(intorder, firstlo, firsthi, secondlo, secondhi,thirdlo, thirdhi); IntVisual(intorder, firstlo, firsthi, secondlo, secondhi,thirdlo, thirdhi,0);

3) For the three regions described in exercise 2 above. plot the region of integration (a) The region bounded by the planes, x=0, y=0, z=0, x+y=1, and y+2z=2. (b) The wedge cut from the cylinder 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 by the planes z=0 and z=-y, with y\342\211\2440. (c) The region bounded by x=0, y=0, z=0, 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, and 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. 4) Either produce a visualization of the regions integrated over below or explain why the integral does not make sense. (a) 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 (b) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiotSShtc3Vic3VwR0YkNictSSNtb0dGJDYvUSgmIzg3NDc7RicvJStmb3JlZ3JvdW5kR1EuWzE0NCwxNDQsMTQ0XUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRJmluZXJ0RicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVC1JI21uR0YkNiRRIjBGJ0Y9LUZYNiRRIjFGJ0Y9LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjJGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRistRiM2Ki1GNDYnRjZGVy1GLDYlUSJ6RicvJSdpdGFsaWNHUSV0cnVlRicvRj5RJ2l0YWxpY0YnRmhuRltvRistRiM2Ki1GNDYnRjZGVy1GLDYlUSJ4RidGZW9GaG9GaG5GW29GKy1GIzYmLUYsNiVRImZGJ0Zlb0Zoby1GNzYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y9RkNGRkZIRkpGTEZORlBGUkZVLUkobWZlbmNlZEdGJDYkLUYjNihGXnAtRjc2LVEiLEYnRj1GQy9GR0Znb0ZIRkpGTEZORlBGUi9GVlEsMC4zMzMzMzMzZW1GJy1GLDYlUSJ5RidGZW9GaG9GXnFGYm9GPUY9Rj1GKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4zZW1GJy8lJmRlcHRoR0Zcci8lKmxpbmVicmVha0dRJWF1dG9GJy1GNzYvUTAmRGlmZmVyZW50aWFsRDtGJ0Y6Rj1GQEZDRkZGSEZKRkxGTkZQRlJGVUZib0Y9RitGZ3FGZXJGZHFGPUYrRmdxRmVyRl5wRj1GK0Y9LUY3Ni1RIjtGJ0Y9RkNGYXFGSEZKRkxGTkZQRlIvRlZRLDAuMjc3Nzc3OGVtRidGPQ== (c) 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

One of the reasons to be able to do the conversion between the limits of integration backwards and forwards is that sometimes we want to change the order of integration. Consider the following problem where we have it set up to integrate with respect to z first, then with respect to y, then with respect to x: intorder := [z,y,x]: firsthi:= 6-x-2*y: firstlo:= 0: secondhi := 3-x/2: secondlo := 0: thirdhi:= 6: thirdlo:= 0: IntCheck(intorder, firstlo, firsthi, secondlo, secondhi,thirdlo, thirdhi); IntVisual(intorder, firstlo, firsthi, secondlo, secondhi,thirdlo, thirdhi,0); This is a 4 sided region with edges x=0, y=0, z=0, and z=6-x-2y. In this case we can integrate in any order. Since we did [z,y,x], lets try [y,x,z]. To integrate with respect to y first, we have top z=6-x-2y and bottom y=0. Solve for y. intorder := [y,x,z]: firsthi:= 3-x/2-z/2: firstlo:= 0: We pick orientation [-90,90] to look down the y axis. The edges of the triangle will be the intersections of surfaces with the front or back. We obviously get x=0 and z=0. The intersection of the back, y=0, with the front, y=3-x/2-z/2 gives 0=3-x/2-z/2. Since we next want to integrate with respect to x, we solve the top and bottom for x. secondhi := 6-z: secondlo := 0: We then look at the shadow on the third axis (z) for the final limits. thirdhi:= 6: thirdlo:= 0: We can now check the integral and the visualization. IntCheck(intorder, firstlo, firsthi, secondlo, secondhi,thirdlo, thirdhi); IntVisual(intorder, firstlo, firsthi, secondlo, secondhi,thirdlo, thirdhi, 0); We see that the new integral covers the same region as the old integral.

5) For the integrals below, change the order of integration to make the problem easier to solve (one you can do by hand) and then evaluate. (a) 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. (b) 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. (c) 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. JSFH

After mastering the mechanics of triple integration it is worthwhile to go back to the theory. For a single integral 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 we in some sense add up the value of f(x) over the segment of the x axis bounded by x=a and x=b. For the double integral 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, for each value of x from x=a to x=b, we evaluate the integral 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. In some sense add up the value of f(x,y) over the collection of line segments, one for each value of x from x=a to x=b, with the segment going from g(x) to h(x). For the triple integral LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkobXN1YnN1cEdGJDYnLUkjbW9HRiQ2L1EmJmludDtGJy8lK2ZvcmVncm91bmRHUS5bMTQ0LDE0NCwxNDRdRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnL0krbXNlbWFudGljc0dGJFEmaW5lcnRGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZMLUYjNiQtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvRjZRJ2l0YWxpY0YnRjUtRiM2JC1GUjYlUSJiRidGVUZYRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMkYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRlI2I1EhRictRiM2Ki1GLDYnRi4tRiM2JS1GUjYlUSJnRidGVUZYLUkobWZlbmNlZEdGJDYkLUYjNiQtRlI2JVEieEYnRlVGWEY1RjVGNS1GIzYlLUZSNiVRImhGJ0ZVRlhGW3BGNUZpbkZcb0Zfby1GIzYqLUYsNidGLi1GIzYlLUZSNiVRImpGJ0ZVRlgtRlxwNiQtRiM2JkZgcC1GLzYtUSIsRidGNUY7L0Y/RldGQEZCRkRGRkZIRkovRk5RLDAuMzMzMzMzM2VtRictRlI2JVEieUYnRlVGWEY1RjVGNS1GIzYlLUZSNiVRImtGJ0ZVRlhGYXFGNUZpbkZcb0Zfby1GIzYmLUZSNiVRImZGJ0ZVRlgtRi82LVEwJkFwcGx5RnVuY3Rpb247RidGNUY7Rj5GQEZCRkRGRkZIRkpGTS1GXHA2JC1GIzYoRmBwRmVxRltyRmVxLUZSNiVRInpGJ0ZVRlhGNUY1RjVGX28tSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuM2VtRicvJSZkZXB0aEdGZ3MvJSpsaW5lYnJlYWtHUSVhdXRvRictRi82L1EwJkRpZmZlcmVudGlhbEQ7RidGMkY1RjhGO0Y+RkBGQkZERkZGSEZKRk1GX3NGNUZfb0Zic0ZgdEZbckY1Rl9vRmJzRmB0RmBwRjU=, for each value of x from x=a to x=b, we evaluate the integral 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. In some sense add up the value of f(x,y, z) over the collection of planar regions, one for each value of x from x=a to x=b.

We first set up a procedure to produce slices for a fixed value of the third variable. Intslice := proc(intorder, firsthi, firstlo, secondhi, secondlo, thirdlo, thirdhi, sval) local third0, second0, firstfront, firstback, first0, maplabels, topbottom, leftright, frontback, slice, lines: third0 := sval*thirdhi+(1-sval)*thirdlo; second0 := t -> t*eval(secondhi,intorder[3]=third0)+ (1-t)*eval(secondlo,intorder[3]=third0); firstfront := t -> eval(firsthi, {intorder[2]=second0(t), intorder[3]=third0}): firstback := t -> eval(firstlo, {intorder[2]=second0(t), intorder[3]=third0}): first0 := (t,u) -> u*firstfront(t)+ (1-u)*firstback(t): maplabels := x -> [convert(x[3],string),convert(x[2],string), convert(x[1],string)]: maplabels(intorder); lines:= spacecurve([seq( [third0, second0(j/10), first0(j/10,u)], j=0..10)],u=0..1); slice := plot3d( [third0, second0(t),first0(t,u)], t=0..1, u=0..1,color=[gold]): display3d({slice,lines}, axes=boxed, style=patchcontour, labels=maplabels(intorder)); end proc: Next first set up the problem, check the integral and the region. intorder := [z,x,y]: firsthi:= sqrt(1-x^2): firstlo:= 0: secondhi := sqrt(1-y^2): secondlo := 0: thirdhi:= 1: thirdlo:= 0: IntCheck(intorder, firstlo, firsthi, secondlo, secondhi,thirdlo, thirdhi); IntVisual(intorder, firstlo, firsthi, secondlo, secondhi,thirdlo, thirdhi,0); Finally we look at an animation of the region in terms of slices display3d([seq(display3d([Intslice(intorder, firstlo, firsthi, secondlo, secondhi,thirdlo, thirdhi,i/10), IntVisual(intorder, firstlo, firsthi, secondlo, secondhi,thirdlo, thirdhi,.9)]),i=0..10)], insequence=true);

JSFH JSFH