Calculus 10.3 day 2 - University of Houston

Calculus 10.3 day 2 - University of Houston

10.3 day 2 Calculus of Polar Curves Lady Bird Johnson Grove, Redwood National Park, California Photo by Vickie Kelly, 2007 Greg Kelly, Hanford High School, Richland, Washington Try graphing this on the TI-89. r 2sin 2.15 0 16 To find the slope of a polar curve: dy dy d dx dx

d d r sin d d r cos d r sin r cos r cos r sin We use the product rule here. To find the slope of a polar curve: dy dy d dx dx d

d r sin d d r cos d r sin r cos r cos r sin dy r sin r cos dx r cos r sin Example: r 1 cos r sin

sin sin 1 cos cos Slope sin cos 1 cos sin 2 2 sin cos cos sin cos sin sin cos 2 2 sin cos cos 2sin cos sin cos 2 cos sin 2 sin Area Inside a Polar Graph:

The length of an arc (in a circle) is given by r. when is given in radians. For a very small , the curve could be approximated by a straight line and the area could be found using the triangle formula: A 1 bh 2 r r d 1 1 2 dA rd r r d 2 2 1 2 dA r d 2 We can use this to find the area inside a polar graph.

1 dA r 2 d 2 1 2 A r d 2 Example: Find the area enclosed by: r 2 1 cos 1 2 0 2 r d 2 1 2 4 1 cos d 0 2 2

2 2 1 2 cos cos d 2 0 2 0 1 cos 2 2 4 cos 2 d 2 2 0

1 cos 2 2 4 cos 2 d 2 2 3 4 cos cos 2 d 0 1 3 4sin sin 2 2 2 0 6 0 6 Notes:

To find the area between curves, subtract: 1 2 2 A R r d 2 Just like finding the areas between Cartesian curves, establish limits of integration where the curves cross. When finding area, negative values of r cancel out: r 2sin 2 Area of one leaf times 4: 2 1 2 A 4 2sin 2 d 2 0 A 2 Area of four leaves:

2 1 2 A 2sin 2 d 2 0 A 2 To find the length of a curve: Remember: ds dx 2 dy 2 For polar graphs: x r cos y r sin If we find derivatives and plug them into the formula, we (eventually) get: 2

dr ds r d d 2 So: Length 2 dr r d d 2

2 dr r d d 2 Length There is also a surface area equation similar to the others we are already familiar with: When rotated about the x-axis: 2 dr S 2 y r d

d 2 2 dr S 2 r sin r d d 2

Recently Viewed Presentations

  • Excerpt from Abraham Lincoln's Second Inaugural Address

    Excerpt from Abraham Lincoln's Second Inaugural Address

    Excerpt from Abraham Lincoln's Second Inaugural Address "With malice toward none; with charity for all… let us strive on to finish the work we are in… to bind up the nation's wounds; to care for him who shall have borne...
  • Cold Weather Camping - Chief Seattle Council

    Cold Weather Camping - Chief Seattle Council

    Cold Weather Camping. PTC 626Oct 15 2016. Instructor: Brian Pratt. Troop 186 Seattle "There's no bad weather, just bad clothing." ... As with other shelters, you want an arched roof, smooth walls, cold well, ideally with entrance below sleeping level.
  • Working in Partnership Department of Rheumatology with Portsmouth

    Working in Partnership Department of Rheumatology with Portsmouth

    amme. Why attend the 2017- Love Your Bone Health Care Professional Conference? The conference * is the only event dedicated to exploring all aspects of Bone Health and Osteoporosis Management in the Portsmouth and Hampshire region. * will feature a...
  • S2 - P1 Incursion - Pilot

    S2 - P1 Incursion - Pilot

    Adopt sterile cockpit concept. Communication impediments in multi-crew cockpit . Define the role of each pilot. CRM training on communication techniques. Manage the cockpit workload. Communication . h. azards and mitigation. Operations. Operational hazards and mitigations.
  • A Midsummer night's dream

    A Midsummer night's dream

    A Midsummer night's dream. Today: Understanding Shakespeare. Understanding the interaction of the characters and how this advances the plot. Identify topics from the story and compare/contrast them to modern day media
  • MOUNTAIN TOP UNIVERSITY (Empowered to Excel) College of ...

    MOUNTAIN TOP UNIVERSITY (Empowered to Excel) College of ...

    The distinction between flows and stocks can be easily understood by comparing the actions of Still Camera (whichrecords position at a point of time) with that of Video Camera (which records position during a period of time). Dependent variable: This...
  • Assembly Language - cs.colostate.edu

    Assembly Language - cs.colostate.edu

    LC-3 has three condition code registers:N -- negativeZ -- zeroP -- positive (greater than zero) ... Memory controller will need to be able to tell what address map to memory and which ones map to IO devices. 8-Interrupt-Driven I/O. External...
  • Physics 106P: Lecture 1 Notes

    Physics 106P: Lecture 1 Notes

    Steel from podium Painter homework problem. Get demo here have man stand at several places calculate force on each support. ... CCW rotation is positive ACT The picture below shows three different ways of using a wrench to loosen a...