Try graphing this on the TI-89.

Slides:

Advertisements

Similar presentations

Polar Coordinates We Live on a Sphere.

Advertisements

P ARAMETRIC AND P OLAR I NTEGRATION. A REA E NCLOSED P ARAMETRICALLY Suppose that the parametric equations x = x(t) and y = y(t) with c  t  d, describe.

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

Equations of Tangent Lines

Copyright © Cengage Learning. All rights reserved.

Section 10.4 – Polar Coordinates and Polar Graphs.

Polar Graphs and Calculus

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.

9.3 Polar Coordinates 9.4 Areas and Lengths in Polar Coordinates.

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.

10.3 Polar Coordinates. One way to give someone directions is to tell them to go three blocks East and five blocks South. Another way to give directions.

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.

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.

10.1 Parametric Equations. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function.

10.2 – 10.3 Parametric Equations. There are times when we need to describe motion (or a curve) that is not a function. We can do this by writing equations.

7.4 Lengths of Curves. 2+x csc x 1 0 If we want to approximate the length of a curve, over a short distance we could measure a straight line. By the.

10.1 Parametric Functions. In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function.

10.3 day 2 Calculus of Polar Curves Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007 Lady Bird Johnson Grove, Redwood National.

In chapter 1, we talked about parametric equations. Parametric equations can be used to describe motion that is not a function. If f and g have derivatives.

10.6B and 10.7 Calculus of Polar Curves.

10. 4 Polar Coordinates and Polar Graphs 10

Chapter 8 – Further Applications of Integration

Graphing Polar Graphs Calc AB- Section10.6A. Symmetry Tests for Polar Graphs 1.Symmetry about the x -axis: If the point lies on the graph, the point ________.

POLAR COORDINATES MIT – Polar Coordinates click PatrickJMT Polar coordinates – the Basics Graphing Polar Curve – Part 1 Graphing Polar Curve – Part 2 Areas.

7.4 Day 2 Surface Area Greg Kelly, Hanford High School, Richland, Washington(Photo not taken by Vickie Kelly)

Arc Length & Surfaces of Revolution (7.4)

Graphing and Equations

Parametric equations Parametric equation: x and y expressed in terms of a parameter t, for example, A curve can be described by parametric equations x=x(t),

Writing Linear Equations

Using Slopes and Intercepts

A moving particle has position

Lecture 31 – Conic Sections

Slope Slope is the steepness of a straight line..

REVIEW 9.1, 9.3, and 9.4 Polar Coordinates and Equations.

Calculus with Parametric Curves

STANDARD FORM OF A LINEAR EQUATION

Copyright © Cengage Learning. All rights reserved.

1.) Set up the integral to find the area of the shaded region

Write an Equation in Slope-Intercept Form

10.6: The Calculus of Polar Curves

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

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

2-4: Tangent Line Review &

Arc Length and Surface Area

Section 12.6 – Area and Arclength in Polar Coordinates

What if we wanted the area enclosed by:

Writing Equations in Slope-Intercept Form

Warm up Find the area of surface formed by revolving the graph of f(x) = 6x3 on the interval [0, 4] about the x-axis.

POLAR COORDINATE SYSTEMS

Graphing Linear Equations

True or False: The exact length of the parametric curve {image} is {image}

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

Copyright © Cengage Learning. All rights reserved.

Learning Target #21 Equations of Circles.

10.2 – Calculus with Parametric Curves

7.4 Lengths of Curves and Surface Area

True or False: The exact length of the parametric curve {image} is {image}

Parametric Functions 10.1 Greg Kelly, Hanford High School, Richland, Washington.

Sherin Stanley, Sophia Versola

Copyright © Cengage Learning. All rights reserved.

Area of a Surface of Revolution

2.5 Basic Differentiation Properties

Presentation transcript:

Try graphing this on the TI-89.

To find the slope of a polar curve: We use the product rule here.

To find the slope of a polar curve:

Example:

Area Inside a Polar Graph: The length of an arc (in a circle) is given by r . q when q is given in radians. For a very small q, the curve could be approximated by a straight line and the area could be found using the triangle formula:

We can use this to find the area inside a polar graph.

Example: Find the area enclosed by:

Notes: To find the area between curves, subtract: 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: Area of one leaf times 4: Area of four leaves:

To find the length of a curve: Remember: For polar graphs: If we find derivatives and plug them into the formula, we (eventually) get: So:

There is also a surface area equation similar to the others we are already familiar with: When rotated about the x-axis: p