# Lecture 26: Spherical coordinates and surface area

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Lesson Summary:

In this lecture, students learn how to use spherical coordinates to represent points in space using distance to the origin and two angles. The lecture explains how to set up problems using these new coordinates and how to find surface area of objects in space. Students are taught how to slice objects in space and integrate using the volume element of rho, phi, and theta. The lecture also covers examples of finding equations for certain shapes, including cones and the xy-plane.

Lesson Description:

Learn how to use spherical coordinates and their notation, how to set up problems using these new coordinates, and how to find surface area of objects in space.

Denis Auroux. 18.02 Multivariable Calculus, Fall 2007. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (Accessed March 21, 2009). License: Creative Commons Attribution-Noncommercial-Share Alike.

• What are spherical coordinates?
• How do you compute triple integrals using spherical coordinates?
• What do the angles phi and theta represent in spherical coordinates?
• What does the length rho represent in spherical coordinates?
• How do you switch between spherical, cylindrical, and rectangular coordinates?
• What does the equation phi = pi/4 or phi = pi/2 look like?
• How do you know which order to write drho, dtheta, dphi when finding a triple integral?
• How do you find the surface area on a sphere of radius a using triple integrals and spherical coordinates?
• How do you find the volume of a portion of the unit sphere above z = 1/square root of 2?
• How can you find the gravitational force exerted by delta M at (x, y, z) on a mass m at the origin?
• #### Staff Review

• Currently 4.0/5 Stars.
Spherical coordinates are introduced as a better way to find triple integrals of certain objects. The notation and the meaning of terms is described. Also, moving between spherical, cylindrical, and rectangular coordinates is explained. Many good examples are shown in this complex, but understandable lecture.