Mastering Polar Coordinates in Double Integrals

1. Chapter 17 Section 17.4 The Double Integral as the Limit of a Riemann Sum; Polar Coordinates

2. Polar Coordinates Any point in the plane can be obtained by knowing r, the distance the point is from the origin and the angle the point makes with the positive x-axis. The values are called the polar coordinates of the point. Conversions: Polar to Rectangular: Rectangular to Polar: y x Polar coordinates are used to describe curves that have complicated rectangular equations and make them simpler (at least from a calculus perspective (i.e. integrating)). Cardioid. Circles centered at the origin. Spiral. 3 leafed rose.

3. Double Integrals and Polar Coordinates A double integral in rectangular coordinates can be switched to a double integral in polar coordinates by replacing all x’s on the integrand by and all y’s in the integrand by . We need to multiply by the integrating factor . Integrating factor (Do not forget!) y Polar Descriptions of Regions The region is no longer thought of as top and bottom or left and right curves. It must be thought of as outer and inner curve between a starting angle and an ending angle . The angles and for a (which is the most common) must continuously bound between the outer and inner curves. x

4. Example Find the integral to the right where is the region in the first quadrant inside the circle , outside the circle and above the line . First we graph and get the curves in polar form to determine the outside and inside curves.

5. Example Find the integral to the right for the region inside the circle and outside the cardioid . First graph the region and find the angles where the curves intersect. and You might be tempted to use the angles and , but this would get the wrong part of the region. Need to use the angles and . and

6. Example Sometimes you might need to convert a rectangular integral to polar like the integral to the right. First graph the region.