XP 18

Posted: August 27th, 2021

Student’s Name

Instructor’s Name

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Date

XP 18

Given the equation

Question 1

Using your calculator/technology to help, sketch the graph of this function on a set of Polar axes over the interval [0, 2Pi].  Mark the orientation of the graph; mark the intercepts and identify them with their Polar coordinates. Also, identify the angle in this interval for which r is undefined.

Polar plot =>

Polar Plots

Question 2

A). Let R be the region swept out by r over the theta-interval, where.   Set up the Polar-form integral whose value represents the Area of R. 

Integration; definite integral

Series expansion at k=0

Indefinite integral

B). Use the formula

                                                                        (CRC Handbook, #366)

            to help write another formula, using variable k, that would represent the Area of    region R, as defined in part A.

New formula;

Thus,

C). Use your formula from part B to find the Area of R when k =

            I)                                     II)                                    III)     

Make small sketches illustrating what R looks like in each case; report the areas rounded to the nearest tenth.

At k =π/4 area bound is;

  however, π= 3.14 and k = π/4

Area =  = 25.10

Figure 1: Sketch varying k from 0 to π/4

At k =π/2 area bound is;

  however, π= 3.14 and k = π/2

Area =  = 54.20

Figure 2: Sketch varying k from 0 to π/2

At k =3π/4 area bound is;

  however, π= 3.14 and k = 3π/4

Area =  = 75.43

Figure 3:Sketch varying k from 0 to 3π/4