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Section 16.8

- Triple Integrals in Cylindrical and Spherical

Coordinates

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).

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.

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

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

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

TRIPLE INTEGRATION IN SPHERICAL COORDINATES

The Riemann sum on the previous slide gives

us where E is a spherical wedge given by

EXTENSION OF THE FORMULA

The formula can be extended to included more

general spherical regions such as The triple

integral would become

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.

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