index_17.html

Double and Triple integrals

When working a triple integral, you sometimes have to work in alternate coordinate systems. You can use some of the techniques from above to make the computations easier.

For example, if you had to do the following integral:

Underscript[∫∫∫, R] dV, where R is the region defined by R={(u,v,φ}|1≤u≤,2≤v≤3, 0≤φ≤} in Paraboloidal coordinates.

First, let's examine the region by graphing each of the boundaries (this is a pain, but I haven't thought of a faster method yet):

In[37]:=

faceULo = ParametricPlot3D[Evaluate[CoordinatesToCartesian[{1, v, ϕ}, Paraboloidal]], {v, 2, 3}, {ϕ, 0, π/6}]

-Graphics3D-

Out[37]=

In[38]:=

$faceUHi = ParametricPlot3D[Evaluate[CoordinatesToCartesian[{v^2, v, ϕ}, Paraboloidal]], {v, 2, 3}, {ϕ, 0, π/6}]$

-Graphics3D-

Out[38]=

In[39]:=

faceVLo = ParametricPlot3D[Evaluate[CoordinatesToCartesian[{u, 2, ϕ}, Paraboloidal]], {u, 1, 9}, {ϕ, 0, π/6}]

$General :: spell1 : Possible spelling error: new symbol name \"faceVLo\" is similar to existing symbol \"faceULo\". More…$

-Graphics3D-

Out[39]=

In[40]:=

faceVHi = ParametricPlot3D[Evaluate[CoordinatesToCartesian[{u, 3, ϕ}, Paraboloidal]], {u, 1, 9}, {ϕ, 0, π/6}]

$General :: spell1 : Possible spelling error: new symbol name \"faceVHi\" is similar to existing symbol \"faceUHi\". More…$

-Graphics3D-

Out[40]=

In[41]:=

faceϕLo = ParametricPlot3D[Evaluate[CoordinatesToCartesian[{u, v, 0}, Paraboloidal]], {u, 1, 9}, {v, 2, 3}]

-Graphics3D-

Out[41]=

In[42]:=

faceϕHi = ParametricPlot3D[Evaluate[CoordinatesToCartesian[{u, v, π/6}, Paraboloidal]], {u, 1, 9}, {v, 2, 3}]

-Graphics3D-

Out[42]=

In[43]:=

-Graphics3D-

Out[43]=

Actually, let's leave off one of the sides (which face is that?):

In[44]:=

-Graphics3D-

Out[44]=

So, not a great candidate for integration in Cartesian coordinates. (It's the region on the inside.) Let's convert everything into Paraboloidal:

$fCartesian[{x_, y_, z_}] := x^3 (x^2 + y^2)^(1/2)$

$fParaboloidal[{u_, v_, ϕ_}] = fCartesian[CoordinatesToCartesian[{u, v, ϕ}, Paraboloidal[u, v, ϕ]]]//FullSimplify$

$General :: spell1 : Possible spelling error: new symbol name \"fParaboloidal\" is similar to existing symbol \"Paraboloidal\". More…$

$General :: spell1 : Possible spelling error: new symbol name \"jDet\" is similar to existing symbol \"Det\". More…$

$∫_0^(π/6) ∫_2^3∫_1^v^2fParaboloidal[{u, v, ϕ}] jDetuvϕ$

Problem

Find Underscript[∫∫∫, R] dV, where R is the region defined by R={(u,v,z}|0≤u≤1,2≤v≤+3, 0≤z≤u v} in ParabolicCylindrical coordinates. Graph the region of integration as well.

Created by Mathematica (April 3, 2007)