We now calculate the volume delimiting both surfaces in space. For this we will use a double integral. To define the boundaries of integrals, we will represent surface projections from different points of view.
We will first calculate the normal ends of the function and then compare them to the conditioned ends. We will calculate the conditioned ends using the Lagrange multiplier method.
Cartesian equations:
sphere x2 + y2 + z2 = Cylinder 4: x2 + y2 - 2y = 0Parametric equations:
Dial: | x = 2cos[t] cos[v], | cylinder: | x = 2sin[t] cops[t], |
and = 2sin[t] cos[v], | and = 2sen[t]2, | ||
z = 2sen[v], | z = z |
Three-dimensional image:
-Graphics3D-
z = ParametricPlot3D[ {2sin[t]cos[t],2sin[t]2,z}, {t,0,2{}, {z,-2,2}, ViewPoint Rule {3,1,1.5}, DisplayFunction> Identity]
-Graphics3D-
Show[e, z, DisplayFunction: $DisplayFunction]
-Graphics3D-
Plant projection:
pc=Graphics[{Circle[{0,0},2], {RGBCoso[1,0,0], Disk[{1,0},1]}, Axes> True, <unk> Ratio Automatic]
-Graphics3D-
Show[xliff-newline]
-Graphics3D-
Previous screening:
Abr=Graphics[{{RGBCsmell[1,0,0], Rectangle[{0,-2},{2,2}]}, Circle[{0,0},2]}, Axes- True, <unk> Ratio-Automatic]
-Graphics3D-
Show[abr]
-Graphics3D-
Projection from the right:
epr=Graphics[{{RGBCsmell[1,0,0], Rectangle[{-1,-2},{1,2}]}, Circle[{0,0},2]}, Axes> True, <unk> Ratio Rule Automatic]
-Graphics3D-
Show[epr, Axes> True]
-Graphics3D-
Double integral:
Limits of integrals in Cartesian coordinates:
and Prudencio [0, 2],x Suitable [-] (2 and - y2), » (2 and - y2)]integrated: z = » (4 - x2 - y2).Limits of integrals in cylindrical coordinates:
integrating
z = r * (4 - r2)Note: with these limits we will only calculate the upper half of the volume.
Integrate[r*Sqrt[4-r2],{t,0,{},{r,0,2sin[t]}]?8/3
+ 4/9 (-4 + 3ess) - 4/9 (4 + 3?)
Simplify[%]
8/9 (-4 + 3ler)
This is the upper half of the volume. Therefore, the total volume:
2 * 16/9
(-4 + 3ash)
First we made a three-dimensional image to capture the idea of volume. For this we have used the traditional ParametricPlot3D and Show commands and have chosen the standard approach through the ViewPoint feature. Next, we have represented the three volume projections, top, front and right, using the Disk, Rectangle, Circle, Graphics and Show commands. In it we have also taken advantage of DisplayFunction, RGBC odor, <unk> Ratio and Axes. Finally, we calculated the double integral with the Integrate order and simplified the result using Simplify. In this way we have achieved half the volume. The total volume is obtained by multiplying the previous result (%) by two.