MTH 253 Calculus (Other Topics) Chapter 10 – Conic Sections and Polar Coordinates Section 10.7 – Areas and Length in Polar Coordinates Copyright © 2009 by Ron Wallace, all rights reserved.

Integrals – A reminder … A definite integral is a limit of a Riemann sum.

Areas in Polar Coordinates  Fan shaped regions bounded by a polar curve and two rays emanating from the pole (i.e. origin).

Areas in Polar Coordinates 1.Divide the region into n pieces with evenly spaced rays emitting from the pole. 2.Approximate the area of each piece using a sector of a circle. 3.Add up all of the areas of the sectors. 4.Take the limit as the number of sectors approaches infinity. 5.Which gives an integral.  

Areas in Polar Coordinates Example: Find the area inside one pedal of the curve

Areas Between Polar Curves  Subtract Regions  

Areas Between Polar Curves  Subtract Regions   Yellow Area:

Areas Between Polar Curves  Subtract Regions Yellow Area:   Pink Area:

Areas Between Polar Curves  Subtract Regions Yellow Area:   Pink Area: Green Area (between the curves) = Yellow Area – Pink Area NOTE: Use either form, whichever make sense to you!

Areas Between Polar Curves Example: Find the area between the curves … Red Area: Total Area:

Lengths of Polar Curves  Curves bounded two rays emanating from the pole (i.e. origin).

Length of a Curve - Review yy xx s (a,b) (c,d) 1 of 2

Length of a Curve - Review 2 of 2 ab m & n are either values of x, values of y, or values of t depending on how the function is defined (explicitly or parametrically).

Lengths of Polar Curves

  Example: Find the length of the spiral …