What if we wanted the area enclosed by:

Slides:

Advertisements

Similar presentations

Consider the function We could make a graph of the slope: slope Now we connect the dots! The resulting curve is a cosine curve. 2.3 Derivatives of Trigonometric.

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.

Area between curves. How would you determine the area between two graphs?

Remember: Derivative=Slope of the Tangent Line.

Trapezium Rule Formula given 5 ordinates means n=4 strips.

Equations of Tangent Lines

Equations of Tangent Lines

Find the slope of the tangent line to the graph of f at the point ( - 1, 10 ). f ( x ) = 6 - 4x

Derivatives - Equation of the Tangent Line Now that we can find the slope of the tangent line of a function at a given point, we need to find the equation.

7.1 Area Between 2 Curves Objective: To calculate the area between 2 curves. Type 1: The top to bottom curve does not change. a b f(x) g(x) *Vertical.

Section 3.5a. Evaluate the limits: Graphically Graphin What happens when you trace close to x = 0? Tabular Support Use TblStart = 0, Tbl = 0.01 What does.

Clear your desk (except for pencil) for the QUIZ.

Volume of a Solid by Cross Section Section 5-9. Let be the region bounded by the graphs of x = y 2 and x=9. Find the volume of the solid that has as its.

Review: Volumes of Revolution. x y A 45 o wedge is cut from a cylinder of radius 3 as shown. Find the volume of the wedge. You could slice this wedge.

Section 10.4 – Polar Coordinates and Polar Graphs.

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...

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.

Every slope is a derivative. Velocity = slope of the tangent line to a position vs. time graph Acceleration = slope of the velocity vs. time graph How.

9.3 Polar Coordinates 9.4 Areas and Lengths in Polar Coordinates.

Section 12.6 – Area and Arclength 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.

10.3 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.

(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.

Areas & Definite Integrals TS: Explicitly assessing information and drawing conclusions.

Let R be the region bounded by the curve y = e x/2, the y-axis and the line y = e. 1)Sketch the region R. Include points of intersection. 2) Find the.

1 An isosceles triangle has one line of symmetry.

The Quick Guide to Calculus. The derivative Derivative A derivative measures how much a function changes for various inputs of that function. It is.

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

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 ________.

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...

Assigned work: pg.83 #2, 4def, 5, 11e, Differential Calculus – rates of change Integral Calculus – area under curves Rates of Change: How fast is.

Print polar coordinates for hw

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

Linear Approximations. In this section we’re going to take a look at an application not of derivatives but of the tangent line to a function. Of course,

The Disk Method (7.2) February 14th, 2017.

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),

Definite integration results in a value.

DIFFERENTIATION RULES.

Warm Up Determine for y2 + xy + 3x = 9.

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

Section 11.3A Introduction to Derivatives

Copyright © Cengage Learning. All rights reserved.

10.6: The Calculus of Polar Curves

Try graphing this on the TI-89.

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

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

The Quick Guide to Calculus

Volume by Cross Sections

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

2-4: Tangent Line Review &

By Kevin Dai, Minho Hyun, David Lu

Section 12.6 – Area and Arclength in Polar Coordinates

Area & Volume Chapter 6.1 & 6.2 February 20, 2007.

PROVING A TRIANGLE IS AN ISOSCELES TRIANGLE

PROVING A TRIANGLE IS AN ISOSCELES TRIANGLE

Copyright © Cengage Learning. All rights reserved.

More with Rules for Differentiation

Sherin Stanley, Sophia Versola

Copyright © Cengage Learning. All rights reserved.

Presentation transcript:

What if we wanted the area enclosed by: Or the slope of this tangent line?

Let’s start with the slope which for any curve is equal to But how does one find this in relation to polar functions? Here’s what we know: How? Product Rule, of course

To find the slope of a polar curve:

Find the slope of this tangent line at

Find the equation of this tangent line at

Find the equation of this tangent line at

What about the area enclosed by: Start with this slice

What about the area enclosed by: This slice is approximately the shape of an isosceles triangle The area requires that we know height and base. Remember arclength from trig? where  must be in radians In this case, the angle is very small so we will call it d So the area of this slice can be approximated as

What about the area enclosed by: To extend this approximation over the whole region we will need an… Integral I knew you wouldn’t forget  and  are usually 0 and 2p but it depends upon the problem

Now find the area… Example: Find the area enclosed by:

The shaded region to the right is bounded by the polar curves: But what about this one? The shaded region to the right is bounded by the polar curves: and Find the area of the shaded region. We need to find  and . So we need to set the two curves equal to each other.

To find the area between curves, subtract: Just like finding the areas between Cartesian curves, establish limits of integration where the curves cross.

Using Symmetry Area of one leaf times 4: Area of four leaves:

p 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: p