Sec. 10.5: Arc Length in Polar Coordinates

1. Sec. 10.5: Arc Length in Polar Coordinates Let f be a function whose derivative is continuous on the interval  ≤  ≤ . The length of the graph ofr = f() from  =  to  =  is Theorem 10.14: Arc Length of a Polar Curve

2. Sec. 10.5: Arc Length in Polar Coordinates Ex: Find the length of one complete arc of The graph makes a complete arc between (Find this by inspection.)

3. AP Calculus BCFriday, 14 February 2014 • OBJECTIVETSW (1) find the area of a region bounded by a polar graph, and (2) find the points of intersection of two polar graphs. • ASSIGNMENTS DUE TUESDAY • Arc Length book work (Sec. 7.4, 10.3) • WS Polar Arc Length • Sec. 10.5 • TEST: Sec. 7.1 – 7.4, 10.3, 10.5is on Tuesday, 18 February 2014 and (part of) Wednesday/Thursday. • The Cake Lab is still on Friday, 21 February 2014.

4. Sec. 10.5: Area in Polar Coordinates

5. Sec. 10.5: Area in Polar Coordinates Instead of using rectangles to find area, we will instead use sectors of circles. What is the area of a sector of a circle? To find the area of a sector, set up a ratio:

6. Sec. 10.5: Area in Polar Coordinates Using the same process and reasoning as was used to find the area under a curve in the rectangular coordinate system, we can find area in the polar coordinate system.

7. Sec. 10.5: Area in Polar Coordinates If f is continuous and nonnegative on the interval [, ], 0 <  –  ≤ 2, then the area of the region bounded by the graph of r = f() between the radial lines  =  and  =  is given by Theorem 10.13: Area in Polar Coordinates

8. Sec. 10.5: Area in Polar Coordinates The key to finding polar area is to determine (by inspection) what makes a "complete" curve. This will indicate the lower and upper limits of integration. You will have to investigate θmin and θmax values in the WINDOW menu to determine the lower and upper limits of integration. It will be helpful to have the "show path" option turned ON in the equation editor.

9. Sec. 10.5: Area in Polar Coordinates Ex: Find the area of one petal of the rose curve This is a 3-petal rose curve. What is the domain for  to make a complete graph? [0, ] We want the area of one petal.

10. Ex: Find the area between the inner and outer loops of the limacon Sec. 10.5: Area in Polar Coordinates We will subtract the area of the inner loop from the area of the outer loop. Crosses the pole when r = 0. Area of inner loop:

11. Ex: Find the area between the inner and outer loops of the limacon Sec. 10.5: Area in Polar Coordinates We will subtract the area of the inner loop from the area of the outer loop. Crosses the pole when r = 0. Area of outer loop: These two values will make a complete outer loop without the inner loop.

12. Ex: Find the area between the inner and outer loops of the limacon Sec. 10.5: Area in Polar Coordinates We will subtract the area of the inner loop from the area of the outer loop. Crosses the pole when r = 0. Area between inner & outer loop:

13. Sec. 10.5: Area in Polar Coordinates Ex: Find the points of intersection of the graphs of r = 1 – 2cos and r = 1. Can we solve simultaneously to find the points? Why wasn’t the third point of intersection found?

14. Sec. 10.5: Area in Polar Coordinates Ex: Find the points of intersection of the graphs of r = 1 – 2cos and r = 1. This third point does not occur with the same coordinates in the two graphs. For the limacon, this third point is (–1, 0). This is why you should always sketch a graph! For the circle, this third point is (1, ).

15. Sec. 10.5: Area in Polar Coordinates Ex: Find the area between and r = −6cosθ: [0, π] r = 2 – 2cosθ: [0, 2π] What are the periods? The curves intersect when their r-values are equal. Since the graphs are symmetrical wrt the horizontal axis, we'll find the area of the upper half and multiply by 2.

16. Sec. 10.5: Area in Polar Coordinates Ex: Find the area between and Half of the area So the area we want is: