# Lecture 17: Double integrals in polar coordinates and applications

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Lesson Summary:

In Lecture 17, the focus is on double integrals in polar coordinates and their application problems. By following the simple approach of slicing the region radially and then integrating in the order of r first and then theta, the double integral can be computed easily in polar coordinates. The lesson also covers the uses of double integrals, such as finding the area of a region and the average value of a function in a region. The applications of double integrals to determine the mass of a flat object with varying density is also discussed thoroughly.

Lesson Description:

Learn more about double integrals, how to find them using polar coordinates, and some application problems involving them.

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.

• How do you find double integrals using polar coordinates?
• What are some applications of double integrals?
• How do you find the double integral of 1 - x^2 - y^2 dydx over the quarter disc region bounded by x^2 + y^2 = 1 in the first quadrant using polar coordinates?
• How do you set up the limits of integration for a double integral using dr dtheta?
• How do you find the mass of a flat object with a given density using double integrals?
• How do you find the average value of a function in a given region using double integrals?
• How do you find the weighted average of a function in a given region with a given density using double integrals?
• How do you find the center of mass of an object using double integrals?
• How do you find the moment of inertia of an object using double integrals?
• What is the moment of inertia of an object?
• How do you find the moment of inertia about the center of a uniform disk of radius a using double integrals?
• How do you find the moment of inertia about a point on the circumference of a uniform disk of radius a using double integrals?
• #### Staff Review

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This lesson shows how some double integrals become much simpler by using polar coordinates rather than rectangular coordinates. How to set up the integrals and how to compute them is explained very well in this lesson. This is a necessary lesson for finding certain double integrals. The meat of this lesson, however, is about applications of double integrals. A bunch of really interesting, important applications are discussed in depth.
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This professor is very well-prepared and easy to understand. Excellent examples with ample explanation. Excellent video series.