Problem 25 in Section 16.3
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- Clemence Booth
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1 Problem 5 in Section 16.3 In[1]:= Recall that in this problem we are studying the pyramid bounded by the planes z 6, y 0, y x 4 and x y z 4. In class we calculated all the vertices of this pyramid. Now I will calculate them in Mathematica eqs z 6, y 0, y x 4, x y z 4 Out[1]= In[]:= Out[]= In[3]:= Out[3]= In[4]:= Out[4]= In[5]:= Out[5]= z 6, y 0, x y 4, x y z 4 Each vertex is at the intersection of three planes. So, I solve three of the four equations and get four vertices. pp1 x, y, z. Solve z 6, y 0, y x 4, x, y, z 1 4, 0, 6 pp x, y, z. Solve z 6, y 0, x y z 4, x, y, z 1 5, 0, 6 pp3 x, y, z. Solve z 6, y x 4, x y z 4, x, y, z 1, 6, 6 pp4 x, y, z. Solve y 0, y x 4, x y z 4, x, y, z 1 4, 0, 1 Plot these points:
![Problem_16_3_5.nb In[6]:= Graphics3D PointSize 0.0, Point pp1, Point pp, Point pp3, Point pp4, Opacity 0.](/docs-images/94/118787629/images/2-3.jpg "3, Polygon pp1, pp, pp3, Polygon pp1, pp, pp4, Polygon pp1, pp4, pp3, Polygon pp4, pp, pp3, Thickness 0.")
2 Problem_16_3_5.nb In[6]:= Graphics3D PointSize 0.0, Point pp1, Point pp, Point pp3, Point pp4, Opacity 0.3, Polygon pp1, pp, pp3, Polygon pp1, pp, pp4, Polygon pp1, pp4, pp3, Polygon pp4, pp, pp3, Thickness 0.01, Cyan, Line pp1, pp, Green, Line pp1, pp3, Blue, Line pp, pp3, Red, Line pp1, pp4, Magenta, Line pp, pp4, Yellow, Line pp3, pp4, Text P 1, pp1, 1, 1, Text P, pp, 1, 1, Text P 3, pp3, 1, 1, Text P 4, pp4, 1, 1, PlotRange 5, 6, 1, 8, 7, 13, Axes True, BoxRatios 11, 9, 0, AxesLabel x, y, z Out[6]= In[7]:= There are 6 lines of interest Cyan lic x, y, z. Solve z 6, y 0, x, y, z 1 Out[7]= In[8]:= x, 0, 6 Green lig x, y, z. Solve z 6, y x 4, x, y, z 1 Out[8]= x, 4 x, 6 Blue
3 Problem_16_3_5.nb 3 In[9]:= lib x, y, z. Solve z 6, x y z 4, x, y, z 1 Out[9]= In[10]:= x, 10 x, 6 Red lir x, y, z. Solve y 0, y x 4, x, y, z 1 Out[10]= In[11]:= 4, 0, z Magenta lim x, y, z. Solve y 0, x y z 4, x, y, z 1 Out[11]= In[1]:= x, 0, 4 x Yellow liy x, y, z. Solve y x 4, x y z 4, x, y, z 1 Out[1]= x, 4 x, 3 x In[13]:= How to integrate? (easier way) How to integrate? Fix y to be y0 and find the corresponding points on Green, Blue and Yellow lines lig Out[13]= In[14]:= Out[14]= In[15]:= Out[15]= In[16]:= Out[16]= In[17]:= Out[17]= In[18]:= Out[18]= x, 4 x, 6 lib x, 10 x, 6 liy x, 4 x, 3 x lig. Solve lig y0, x 1 4 y0, y0, 6 lib. Solve lib y0, x 1 10 y0, y0, 6 liy. Solve liy y0, x 1 4 y0, y0, 3 4 y0
4 4 Problem_16_3_5.nb In[19]:= 10 y0 y0.5; ppg 4 y0, y0, 6 ; ppb, y0, 6 ; ppy 4 y0, y0, 3 4 y0 ; Graphics3D PointSize 0.0, Point pp1, Point pp, Point pp3, Point pp4, Opacity 0.3, Polygon pp1, pp, pp3, Polygon pp1, pp, pp4, Polygon pp1, pp4, pp3, Polygon pp4, pp, pp3, Thickness 0.01, Cyan, Line pp1, pp, Green, Line pp1, pp3, Blue, Line pp, pp3, Red, Line pp1, pp4, Magenta, Line pp, pp4, Yellow, Line pp3, pp4, Text P 1, pp1, 1, 1, Text P, pp, 1, 1, Text P 3, pp3, 1, 1, Text P 4, pp4, 1, 1, Polygon ppg, ppb, ppy, PlotRange 5, 6, 1, 8, 7, 13, Axes True, BoxRatios 11, 9, 0, AxesLabel x, y, z Out[0]=
5 Problem_16_3_5.nb 5 In[1]:= Clear y0 ; Manipulate ppg 4 y0, y0, 6 ; 10 y0 ppb, y0, 6 ; ppy 4 y0, y0, 3 4 y0 ; Graphics3D PointSize 0.0, Point pp1, Point pp, Point pp3, Point pp4, Opacity 0.3, Polygon pp1, pp, pp3, Polygon pp1, pp, pp4, Polygon pp1, pp4, pp3, Polygon pp4, pp, pp3, Thickness 0.01, Cyan, Line pp1, pp, Green, Line pp1, pp3, Blue, Line pp, pp3, Red, Line pp1, pp4, Magenta, Line pp, pp4, Yellow, Line pp3, pp4, Text P 1, pp1, 1, 1, Text P, pp, 1, 1, Text P 3, pp3, 1, 1, Text P 4, pp4, 1, 1, Polygon ppg, ppb, ppy, PlotRange 5, 6, 1, 8, 7, 13, Axes True, BoxRatios 11, 9, 0, AxesLabel x, y, z, y0,, 0, 6 y0 Out[1]=
6 6 Problem_16_3_5.nb In[]:= Out[]= 16 In[3]:= Integrate Integrate Integrate 1, z, 6, 4 x y, x, y 4, 10 y, y, 0, 6 How to integrate? (harder way) How to integrate? Fix z to be z0 and find the corresponding points on Red, Magenta and Yellow lines lir Out[3]= In[4]:= Out[4]= In[5]:= Out[5]= In[6]:= Out[6]= In[7]:= Out[7]= In[8]:= Out[8]= 4, 0, z lim x, 0, 4 x liy x, 4 x, 3 x lir. Solve lir 3 z0, z 1 4, 0, z0 lim. Solve lim 3 z0, x 1 4 z0, 0, z0 liy. Solve liy 3 z0, x 1 z0 3, 4 z0 3, z0
![Problem_16_3_5.nb 7 In[9]:= Clear z0 ; Manipulate ppr 4, 0, z0 ; ppm 4 z0, 0, z0 ; ppy1 z0 3, 4 z0 3, z0 ; Graphics3D PointSize 0.0, Point pp1, Point pp, Point pp3, Point pp4, Opacity 0.](/docs-images/94/118787629/images/7-2.jpg "3, Polygon pp1, pp, pp3, Polygon pp1, pp, pp4, Polygon pp1, pp4, pp3, Polygon pp4, pp, pp3, Thickness 0.")
7 Problem_16_3_5.nb 7 In[9]:= Clear z0 ; Manipulate ppr 4, 0, z0 ; ppm 4 z0, 0, z0 ; ppy1 z0 3, 4 z0 3, z0 ; Graphics3D PointSize 0.0, Point pp1, Point pp, Point pp3, Point pp4, Opacity 0.3, Polygon pp1, pp, pp3, Polygon pp1, pp, pp4, Polygon pp1, pp4, pp3, Polygon pp4, pp, pp3, Thickness 0.01, Cyan, Line pp1, pp, Green, Line pp1, pp3, Blue, Line pp, pp3, Red, Line pp1, pp4, Magenta, Line pp, pp4, Yellow, Line pp3, pp4, Text P 1, pp1, 1, 1, Text P, pp, 1, 1, Text P 3, pp3, 1, 1, Text P 4, pp4, 1, 1, Thickness 0.01, Cyan, Line ppr, ppm, Green, Line ppr, ppy1, Blue, Line ppm, ppy1, PointSize 0.0, Point ppr, Point ppm, Point ppy1, Polygon ppr, ppm, ppy1, PlotRange 5, 6, 1, 8, 7, 13, Axes True, BoxRatios 11, 9, 0, AxesLabel x, y, z, z0,, 6, 1 z0 Out[9]=
8 8 Problem_16_3_5.nb In[30]:= Out[30]= Recall ppy1 z0 z0 3, 4 z0 3, z0 3, 4 z0 3, z0 So, y is between 0 and 4 z 3. In[31]:= Recall, the green and blue line, but now change them to be at the level z0 Green ligz x, y, z. Solve z z0, y x 4, x, y, z 1 Out[31]= In[3]:= x, 4 x, z0 Blue libz x, y, z. Solve z z0, x y z 4, x, y, z 1 Out[3]= x, 4 x z0, z0 In[33]:= Out[33]= 16 Integrate Integrate Integrate 1, x, y 4, 4 y z, y, 0, 4 z, z, 6, 1 3
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