MAT 1236 Calculus III Section 15.8, 15.9 Triple Integrals in Cylindrical and Spherical Coordinates

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Preview Double integrals: For regions related to circles, double integrals are better done in polar coordinates (PC) than rectangular coordinates (RC). Common regions for Triple integrals: (3D) regions related to cylinders (3D) regions related to sphere

Recall: 15.7 Example 3 Find the volume of the solid bounded by the paraboloids

Cylindrical Coordinates Replace the first 2 RC by PC Direct extension of the PC. No new formula are needed. We used CC in 15.7 Example 3.

Example 1 Evaluate where E is the solid lies between, above the xy-plane and below the plane

Example 1

Evaluate where E is the solid lies between, above the xy-plane and below the plane

Spherical Coordinates

New – More Examples 74/readings/sphcoord/ 74/readings/sphcoord/ Variable = constants

Example 2 (a)

Example 2 (b)

Example 2 (c)

Spherical Wedge

Example 3 Evaluate where H is the region lies above the xy-plane and below

Remarks One may attempt to use CC for the integral in example 3.

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