Section 8.3 Area and Arc Length in Polar Coordinates

Review of polar coordinates Any point in the place can be represented as rectangular coordinates or polar coordinates Where rectangular coordinates require the horizontal and vertical distance from the origin, polar coordinates require the distance from the origin and the angle between the ray the point lies on and the positive side of the horizontal axis. Let’s take a look

Suppose P is the indicated point in the xy-plane Then What are the rectangular coordinates given by (5,π/6)? What are the polar coordinates given by (-4, 5). P(x,y) = P(r,θ) x y r θ

Graphs of Equations in Polar Coordinates Some graphs are easier to represent in polar coordinates –What does the graph r = 2 look like? –What does the graph r = θ look like? Let’s take a look at these and some other graphs using Maple –Note: You can graph these in your graphing calculator as well by switching the mode to polar Now let’s talk about calculating the area enclosed by a polar graph

Recall the area of a sector Imagine the area of our polar graph being broken up into infinitely small sectors Let’s find the area of a four leaved rose

Let’s find the area inside r = 3cosθ and outside r = 1 + cosθ Here is a plot

Arc Length in Polar Coordinates Recall our Arc Length formula We will use this to derive the Arc Length formula for polar coordinates And we get