Triple Integrals in Cylindrical and Spherical Coordinates

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Section 16.8

• Triple Integrals in Cylindrical and Spherical
  Coordinates

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CYLINDRICAL COORDINATES
Recall that Cartesian and Cylindrical coordinates
are related by the formulas x r cos ?, y r
sin ?, x2 y2 r2. As a result, the function
f (x, y, z) transforms into f (x, y, z) f (r
cos ?, r sin ?, z) F(r, ?, z).
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TRIPLE INTEGRATION WITH CYLINDRICAL COORDINATES
Let E be a type 1 region and suppose that its
projection D in the xy-plane can be described
by D (r, ?) a ? ß, h1(?) ? h2(?) If
f is continuous, then
NOTE The dz dy dx of Cartesian coordinates
becomes r dz dr d? in cylindrical coordinates.
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EXAMPLES
1. Find the mass of the ellipsoid E given by
4x2  4y2 z2 16, lying above the xy-plane.
Then density at a point in the solid is
proportional to the distance between the point
and the xy-plane. 2. Evaluate the integral
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SPHERICAL COORDINATES AND SPHERICAL WEDGES
The equations that relate spherical coordinates
to Cartesian coordinates are
In spherical coordinates, the counterpart of a
rectangular box is a spherical wedge
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Divide E into smaller spherical wedges Eijk by
means of equally spaced spheres ? ?i,
half-planes ? ?j, and half-cones f fk.
Each Eijk is approximately a rectangular box with
dimensions ??, ?i?f, and ?i sin fk ??. So, the
approximate volume of Eijk is given by Then the
triple integral over E can be given by the
Riemann sum
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TRIPLE INTEGRATION IN SPHERICAL COORDINATES
The Riemann sum on the previous slide gives
us where E is a spherical wedge given by
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EXTENSION OF THE FORMULA
The formula can be extended to included more
general spherical regions such as The triple
integral would become
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EXAMPLES
1. Use spherical coordinates to evaluate the
integral 2. Find the volume of the solid region
E bounded by below by the cone
and above by the sphere x2  y2  z2 3z.