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VC.07. Triple Integrals. Example 1: Triple Integrals to Compute Volume. Example 2: Triple Integrals to Compute Volume. Example 2: Triple Integrals to Compute Volume. Example 3: Changing the Order of Integration. Example 4: 3D-Change of Variables. The Volume Conversion Factor:. - PowerPoint PPT Presentation
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VC.07
Triple Integrals
Example 1: Triple Integrals to Compute Volume
I R
Recall that in previous chapters we could find the length of an interval Iby computing dx or the area of a region R by computing dA.
R
It follows that we can compute the volume of a 3-dimensional region Rby calculating 1dV.For this cube, the calculation is not exciting:
1 x 1,0 y 2,1 z 3
3 2 1
1 0 11dx dy dz
3 2x 1
x 11 0
x dy dz
3 y 2
y 01
2y dz
z 3
z 14z
8
Example 2: Triple Integrals to Compute Volume
We can spice things up a bit. We can findthe volume of the region abovethexy-plane, the xz-plane, the yz-plane, andbelow the plane given by x y 4z 8:
If x y 4z 8, thenz . So z-t8 x y4
8 x
op
is andz-bottomy4 is .0
88 x
x y
0
8
00
4d1 yd dxz
If x y 4z 8, thenwhen z 0, y . S
8 x8 xo y-top is and y-bottom s .0 i
y
x
Finally, integrate x from x = to x 0 = 8.
Example 2: Triple Integrals to Compute Volume
We can spice things up a bit. We can findthe volume of the region abovethexy-plane, the xz-plane, the yz-plane, andbelow the plane given by x y 4z 8:
88 x
x y
0
8
00
4d1 yd dxz
0
8
0
8 x
dy dx1 (8 x y)4
y 8 x
y
8
0 0
2y1 8y xy4 2 dx
28
0
1 (8 x)8(8 x) x(8 x)4 2 dx
643
pyramid base1V A h31 32 23643
g g
Example 3: Changing the Order of Integration
RRepeat Example 2 by computing dy dx dz whereR istheregion
abovethexy-plane, the xz-plane, the yz-plane, andbelow the plane given by x y 4z 8
If x y 4z 8, theny . So y-t8 x 4z8 x 4
op is and y-bottom is .z 0
8 8 x 4z
0
4z2
0 0dzxdy1 d
If x y 4z 8, thenwhen y 0, x . S
8 4z8 4z 0o x-top is and x-bottom is .
xFinally, integrate z from z = to z 0 = 2.
z
643
Example 4: 3D-Change of VariablesxyzCompute the volume of the parallelepiped describedby the region R
between the planes z 3x andz 3x 2,y x and y x 4,and y 2xand y 2x 3.
u z 3x,andleturunfrom0to 2v y x,andlet v runfrom0to 4w y 2x,andlet wrunfrom0to3
xyz uvw
xyzR R
1dx dy dz V (u,v,w) dudv dw
First, pick a clever change of variables:
Our integral is much more manageable now:
3 4 2
xyz0 0 0
V (u,v,w) dudv dw
xyz
yx zu u u
y yxV (u,v,w) v v vyx z
w w w
The Volume Conversion Factor:
xyz uvw
xyzR R
1dx dy dz V (u,v,w) dudv dw
1 2 3
31 2x
31 2
yz
31 2
Let T(u,v,w) be a transformation from uvw-space to xyz-space.That is, T(u,v,w) T (u,v,w),T (u,v,w),T (u,v,w) (x(u,v,w),y(u,v,w),z(u,v,w)).
ThenV (u,v,
TT Tw w w
w) TT Tv
TT Tu u u
v v
xy xyzNote: Just likeA (u,v),weneed V (u,v,w) tobepositive.Hence, the absolutevaluebars in the formula above.
xyz
yx zu u u
yx zvOr V (u,v,w
yv v
xw
)
zw w
Example 4: 3D-Change of VariablesxyzCompute the volume of the parallelepiped describedby the region R
between the planes z 3x andz 3x 2,y x and y x 4,and y 2xand y 2x 3.
u z 3xv y xw y 2x
w vx 32v wy 3
z u w v
xyz
0 0 11 2V (u,v,w) 13 3
1 1 13 3
1 23 31 1 1
3 3
13
1Use .3
3 4 2
xyz0 0 0
V (u,v,w) dudv dw3 4 2
0 0 0
1 dudv dw3
8
Example 4: 3D-Change of VariablesxyzCompute the volume of the parallelepiped describedby the region R
between the planes z 3x andz 3x 2,y x and y x 4,and y 2xand y 2x 3.
parallelepiped
parallelepiped
Check :For a parallelepiped generated by three intersecting vectors X, Y and Z,V (X Y) Z
4 8Parallelepiped is generatedby vectors (1,1,3), , , 4 ,and(0,0,2).3 3
V (1,1,3)
g
4 8, , 4 (0,0,2) 83 3
g
No need to study this, it's just nice to verify this works...
Example 5: A Mathematica-Assisted Change of Variables
2
Find the volume of the "football" whose outer skin is described by the pa 0,0,4s) 4 s cos(t),rametric equation sin(t),0 for2 s
F(s2and0 t
t2
, ) (.
First,fill in the football: 20,0,4s) r 4 s cos(t),sin(t),0
2 sF(r
20 t 20
,s
r
,t) (
1
Change of variables:
2
2
x(r,s,t) r cos(t)y(r,s,t) r sin(t)z(r,
4
s
s
,t) 4s4 s
2 2 1
rst0 2 0
V (r,s,t) dr dsdt
Example 5: A Mathematica-Assisted Change of Variables
2Find the volume of the "football" whose outer skin is described by the param 0,0,4s) (4 s ) cos(t),sinetric equation F(s,t (t),0 for2 s 2and0 t
(2 .
)
xyz
yx zr r r
y yxV (r,s,t) s s syx z
t t t
2 2 1
rst0 2 0
V (r,s,t) dr dsdt
2 44(16r 8rs rs ) (Mathematica)
2048 (Mathematica)15
Example 5: A Mathematica-Assisted Change of Variables
2Find the volume of the "football" whose outer skin is described by the param 0,0,4s) (4 s ) cos(t),sinetric equation F(s,t (t),0 for2 s 2and0 t
(2 .
)
228
8
x4 dx4
204815
Check using solids of revolution:
Example 6: Beyond Volume Calculations
R
R
It is easy to see that dV computes the volume of a solid.But it's
harder to interpret f(x,y,z)dV since f(x,y,z) lives in 4 dimensions.
We need an example to keep referring back to to give us some
intuition.
1 x 1,0 y 2,1 z 3
32 4 g
cm
A cube of varying density has its density at each point (x,y,z) described by f(x,y,z) x y .Findthemassof the cube.
3 2 12 4
1 0 1x y dx dy dz
128 grams15
3D Integrals:
xyz uvw
xyzR R
f(x,y,z)dx dy dz f(x,y,z) V (u,v,w) dudv dw
xyz
yx zu
yx zw
V (u,v,u u
w) yx
w
v
w
zv v
xyzR
xyz
In my opinion, a good way to think about f(x,y,z)dx dy dz is
asacalculation of the mass of R where f(x,y,z) is the density of the solid at any given point (x,y,z).
Example 7: A 3D-Change of Variables with an Integrand
xyz
xyzR
Compute 9y dx dy dz whereR is the parallelepiped that is between
the planes z 3x andz 3x 2, y x and y x 4,and y 2x andy 2x 3.
0 u 20 v 40 w 3
3 4 2
xyz0 0 0
9y(u,v,w) V (u,v,w) dudv dw
u z 3xv y xw y 2x
w vx 32v wy 3
z u w v
xyz1V (u,v,w) 3
xyzR9y dx dy dz
All from Example4:
3 4 2
0 0 0
2v w 19 dudv dw3 3
3 4 2
0 0 02v w dudv dw 132