Area Between Polar Curves

Slides:

Advertisements

Similar presentations

Area in Polar Coordinates

Advertisements

Graphs of Polar Coordinates Sections 6.4. Objectives Use point plotting to graph polar equations. Use symmetry to graph polar equations.

10.2 Polar Equations and Graphs

Integration in polar coordinates involves finding not the area underneath a curve but, rather, the area of a sector bounded by a curve. Consider the region.

Area in Polar Coordinates Lesson Area of a Sector of a Circle Given a circle with radius = r  Sector of the circle with angle = θ The area of.

10.3 Polar Functions Quick Review 5.Find dy / dx. 6.Find the slope of the curve at t = 2. 7.Find the points on the curve where the slope is zero. 8.Find.

Section 12.6 – Area and Arclength in Polar Coordinates

Section 11.4 Areas and Lengths in Polar Coordinates.

AREA BOUNDED BY A POLAR CURVE. The area of the region bounded by a polar curve, r = f (θ ) and the lines θ = α and θ = β is given by: β α A = 1 2 r 2.

10.3 Polar Coordinates. Converting Polar to Rectangular Use the polar-rectangular conversion formulas to show that the polar graph of r = 4 sin.

1 © 2011 Pearson Education, Inc. All rights reserved 1 © 2010 Pearson Education, Inc. All rights reserved © 2011 Pearson Education, Inc. All rights reserved.

Calculus with Polar Coordinates Ex. Find all points of intersection of r = 1 – 2cos θ and r = 1.

10.8 Polar Equations and Graphs. An equation whose variables are polar coordinates is called a polar equation. The graph of a polar equation consists.

Warm Up Calculator Active The curve given can be described by the equation r = θ + sin(2θ) for 0 < θ < π, where r is measured in meters and θ is measured.

A REA AND A RC L ENGTH IN P OLAR C OORDINATES Section 10-5.

Section 16.3 Triple Integrals. A continuous function of 3 variable can be integrated over a solid region, W, in 3-space just as a function of two variables.

Section 10.8 Notes. In previous math courses as well as Pre-Calculus you have learned how to graph on the rectangular coordinate system. You first learned.

Section 15.3 Area and Definite Integral

Sullivan Algebra and Trigonometry: Section 9.2 Polar Equations and Graphs Objectives of this Section Graph and Identify Polar Equations by Converting to.

Section 5.2 – Polar Equations and Graphs. An equation whose variables are polar coordinates is called a polar equation. The graph of a polar equation.

Copyright © 2013, 2009, 2005 Pearson Education, Inc Complex Numbers, Polar Equations, and Parametric Equations Copyright © 2013, 2009, 2005 Pearson.

Section The Polar Coordinate System The keys……

Area of a Region Between Two Curves

10. 4 Polar Coordinates and Polar Graphs 10

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

Double Integrals in Polar Coordinates. Sometimes equations and regions are expressed more simply in polar rather than rectangular coordinates. Recall:

14.3 Day 2 change of variables

Polar Coordinates Lesson 6.3. Points on a Plane Rectangular coordinate system  Represent a point by two distances from the origin  Horizontal dist,

Polar Equations and Graphs. 1. Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation (Similar.

Section The Polar Coordinate System.

Turn in your homework and clear your desk for the QUIZ.

Give 4 different pairs of polar coordinates that correspond to the Cartesian Coordinates (2,2) Aim: How do we describe Curves in Polar form?

Ch. 11 – Parametric, Vector, and Polar Functions 11.3 – Polar Functions.

9.3 - Circles Objectives: Write an equation for a circle given sufficient information. Given an equation of a circle, graph it and label the radius and.

Area between Polar Curves

Polar Coordinates and Graphs of Polar Equations

Graphs of Polar Equations

Polar Coordinates r Pole Polar axis.

Section 6.4 Graphs of Polar Equations

Parametric Equations and Polar Coordinates

Area Bounded by Curves sin x cos x.

5.4 Graphs of Polar Equations

Sketch the region enclosed by {image} and {image}

Sketch the region enclosed by {image} and {image}

8.2 Polar Equations and Graphs

Find the area of the region enclosed by one loop of the curve. {image}

Area in Polar Coordinates

10.3: Polar functions* Learning Goals: Graph using polar form

Polar Coordinates and Polar Graphs

The Natural Logarithmic Function: Integration (Section 5-2)

Area Between Two Curves

Section 12.6 – Area and Arclength in Polar Coordinates

x = 4y - 4 x = 4y + 4 x = 4y - 1 y = 4x - 4 y = 4x - 1 y = 4x + 4

Volumes of Solids of Revolution

Transformations of curves

Polar Coordinates and Graphs of Polar Equations

Math 265 Sections 13.1 – 13.5 Created by Educational Technology Network

10 Conics, Parametric Equations, and Polar Coordinates

HW # −17 , ,20 , ,46 , Row 1 Do Now Test for symmetry with respect to the line

Graphing Polar Equations

Definition A Polar Coordinate System is a method of locating a point (r, ) in a plane where r is a distance from a point called the pole directed at.

Integrals in Polar Coordinates.

Section 6.4 Graphs of Polar Equations

Conics, Parametric Equations, and Polar Coordinates

Polar Coordinates and Graphs of Polar Equations

13.3 Polar Coordinates Rita Korsunsky.

Definite Integrals 5.6 Area Between Curves.

Area and Arc Length in Polar Coordinates

GRAPHS OF THE POLAR EQUATIONS r = a ± b cos θ r = a ± b sin θ

Presentation transcript:

Area Between Polar Curves Section 10.5

Rectangular Coordinates Sketch the area bound by the lines 𝑦=𝑥 and 𝑦=−𝑥+6. Set up an integral to find the area of the region bound by the 2 graphs and the x-axis.

Polar Coordinates Sketch the graphs 𝑟=−6 cos 𝜃 and 𝑟=2−2 cos 𝜃 . Find the area of the region common to the two regions bounded by the curves.

Rectangular Coordinates Sketch the graphs of 𝑦= 𝑥 2 and 𝑦=𝑥. Find the area of the region bounded by these curves.

Polar Coordinates Sketch the curves 𝑟=1 and 𝑟=1− cos 𝜃 Find the area of the region that lies inside the circle and outside the cardioid.

Examples Find the area of the common interior of 𝑟=3 and 𝑟=6 cos (2𝜃) Find the area inside the circle but outside the petal graph. Find the area inside the petal graph but outside the circle.