Section 12.6 – Area and Arclength in Polar Coordinates

Slides:

Advertisements

Similar presentations

Section 7.1 – Area of a Region Between Two Curves

Advertisements

Arc Length Cartesian, Parametric, and Polar. Arc Length x k-1 xkxk Green line = If we do this over and over from every x k—1 to any x k, we get.

PARAMETRIC EQUATIONS AND POLAR COORDINATES

Parametric Equations t x y

 A k = area of k th rectangle,  f(c k ) – g(c k ) = height,  x k = width. 6.1 Area between two curves.

Sec 5 Symmetry in polar coordinate. Definitions Symmetry about the Polar Axis The curve is symmetric about the polar axis ( the x-axis)if replacing the.

Definite Integration and Areas 01 It can be used to find an area bounded, in part, by a curve e.g. gives the area shaded on the graph The limits of integration...

Warm Up Show all definite integrals!!!!! 1)Calculator Active: Let R be the region bounded by the graph of y = ln x and the line y = x – 2. Find the area.

Polar Graphs and Calculus

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.

MTH 253 Calculus (Other Topics) Chapter 10 – Conic Sections and Polar Coordinates Section 10.7 – Areas and Length in Polar Coordinates Copyright © 2009.

Section 12.6 – Area and Arclength in Polar Coordinates

Copyright © Cengage Learning. All rights reserved. 10 Parametric Equations and Polar Coordinates.

Section 8.3 Area and Arc Length in Polar Coordinates.

Section 11.4 Areas and Lengths in Polar Coordinates.

10 Conics, Parametric Equations, and Polar Coordinates

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

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

10.5 Area and Arc Length in Polar Coordinates Miss Battaglia AP Calculus.

Section 10.5 – Area and Arc Length in Polar Coordinates

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.

MA Day 39 – March 1, 2013 Section 12.4: Double Integrals in Polar Coordinates.

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

Sector of a Circle Section  Sector – portion of the area of a circle area of sector = arc measure area of circle 360 Definitions & Formulas.

(r,  ). You are familiar with plotting with a rectangular coordinate system. We are going to look at a new coordinate system called the polar coordinate.

10. 4 Polar Coordinates and Polar Graphs 10

MAT 1226 Calculus II Section 10.4 Areas and Length in Polar Coordinates

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

14.3 Day 2 change of variables

Definite Integration and Areas 01 It can be used to find an area bounded, in part, by a curve e.g. gives the area shaded on the graph The limits of integration...

Clear your desk for the Quiz. Arc Length & Area Arc Length The length of a continuous curve r(θ) on the interval [  ] is equal to.

Sector of a Circle Section  If a circle has a radius of 2 inches, then what is its circumference?  What is the length of the arc 172 o around.

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

Section 10.5 Angles in Circles.

MTH1170 Integrals as Area.

Introduction to Trigonometry

A moving particle has position

Definite integration results in a value.

Lesson 10.6 – Secants, Tangents, and Angle Measure

Parametric Equations and Polar Coordinates

Copyright © Cengage Learning. All rights reserved.

Trigonometric Functions

Scalars and Vectors.

10.6: The Calculus of Polar Curves

Try graphing this on the TI-89.

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

Cross Sections Section 7.2.

Volume by Cross Sections

Area in Polar Coordinates

POLAR COORDINATES Point P is defined by : P .

Polar Area Day 2 Section 10.5A Calculus BC AP/Dual, Revised ©2018

Area Between Polar Curves

Coordinate Geometry in the (x,y) plane.

What if we wanted the area enclosed by:

POLAR CURVES Intersections.

Clicker Question 1 What are two polar coordinates for the point whose rectangular coordinates are (3, 1)? A. (3 + 1, 0) and (-3 – 1, ) B. (3 + 1,

Add or Subtract? x =.

Volume - The Disk Method

Section 7.1 Day 1-2 Area of a Region Between Two Curves

Polar Area Day 3 Section 10.5B Calculus BC AP/Dual, Revised ©2018

Integrals in Polar Coordinates.

Find the area of the shaded region. {image} {image}

Calculus BC AP/Dual, Revised ©2018

Copyright © Cengage Learning. All rights reserved.

Conics, Parametric Equations, and Polar Coordinates

AP Calculus AB 8.2 Areas in the Plane.

Copyright © Cengage Learning. All rights reserved.

Area and Arc Length in Polar Coordinates

Presentation transcript:

Section 12.6 – Area and Arclength in Polar Coordinates 12.2

Integrals in Polar Coordinates Area of a Polar Curve: These are needed to make some integration possible.

The total area of the region enclosed NO CALCULATOR by the polar graph of is

NO CALCULATOR Not an option. Let’s check the graph. Symmetric about the x-axis. So...

CALCULATOR REQUIRED The area of the region enclosed by the graph of the polar curve is 4.712 9.424 18.849 37.699 75.398

CALCULATOR REQUIRED The approximate total area of the region enclosed by the polar graph of is: 0.393 0.785 1.178 1.571 1.873

Find the area inside . It’s a circle! Using calculus: Wait! What?!? The circle actually gets drawn twice. We need to change our limits of integration. Better.

CALCULATOR REQUIRED Set up to definite integral to find the area inside the smaller loop of

Or we could restrict it to exclude the loop. If would have wanted the total area, we would have to change things a bit. Total Area. Inner Loop. Without the loop. We could take the whole area and subtract the inner loop (no double counting). Or we could restrict it to exclude the loop.

CALCULATOR REQUIRED Find the area inside What?!? Let’s try another way.

NO CALCULATOR: Find the area of the region inside the circle r = 4 and outside

We’ll need to subtract red from blue. =7.653

=0.571 We’ll need to add red and blue because they make up two halves. However, the angle ranges are different for the two functions. =0.571

Length of an Arc in Polar Coordinates

CALCULATOR REQUIRED

NO CALCULATOR