14.6 triple integrals in cylindrical and spherical coordinates
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Transcript of 14.6 triple integrals in cylindrical and spherical coordinates
KKKQ1124 ENGINEERING MATHEMATICS (VECTOR CALCULUS)
MS. IZAMARLINA ASSHAARI
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MULTIPLE INTEGRALS Triple Integrals in Cylindrical Coordinates
Useful for circle-symmetrical integration regions and integrand functions
cos( , , ) ( , , )
sin
x rf x y z f r z
y rdxdydz r dr d dz
z z
Switch to polar coordinates for 2 of the 3 coordinates, leave the third as is
Equivalent to integrate first in then in polar coordinates on the projection to the -plane
,zxy
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MULTIPLE INTEGRALS
Example 1
Convert the point to rectangular
coordinates.
5, , 4, ,3
6r z
Example 2
Convert the point to cylindrical coordinates.
, , 1, 3,2x y z
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MULTIPLE INTEGRALS
Example 3
Find an equation in cylindrical coordinates for the surface represented
2 2 .z x y
Example 4
Find an equation in rectangular coordinates for the surface represented by
3sec .r
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MULTIPLE INTEGRALS
Example 5
Using cylindrical coordinates, evaluate
2 2 2 2 2
2
0 0 0; 0 .
a a x a x yx dz d y dx a
Example 6
Find the volume of the solid that is bounded above
and below by the sphere and inside the cylinder
2 2 2 9x y z 2 2 4.x y
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MULTIPLE INTEGRALS Triple Integrals in Spherical Coordinates
Switch to spherical coordinates: radius, longitude, latitude
2 2 2 2
sin cos
sin sin
cos
x
y
z
x y z
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MULTIPLE INTEGRALS
Switch to rectangular coordinates
2 2 2
1
2 2 2
1
cos
tan
x y z
z
x y z
y
x
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MULTIPLE INTEGRALS
Example 7
Convert to rectangular
coordinates and cylindrical coordinates.
2 5, , 2, ,
3 6
Example 8
If the rectangular coordinates of point P are
find the spherical coordinates of P. 1, 3, 2
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MULTIPLE INTEGRALS
Example 9
Find an equation in spherical coordinates for the surface represented by the equation
2 2 2.x y z
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MULTIPLE INTEGRALS
2 sindV d d d
A typical triple integral in spherical coordinates has the form
2 2
1 1
, 2
,
, ,
, , sin
G
h g
h g
f x y z dV
f d d d
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MULTIPLE INTEGRALS
Example 10
Use spherical coordinates to find the volume of the
solid enclosed by the sphere and
the plane
2 2 2 24x y z a
0.z
Example 11
Find the volume of the solid region Q bounded by
the cone and the sphere
2 2z x y 2 2 2 2 .x y z z
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MULTIPLE INTEGRALS
Example 12
Use spherical coordinates to evaluate2 2 2
2 2
2 4 8 2
0 0.
y x y
x yz dz dx d y