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