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Chapter 9: Q. 47 (page 756)

Use Theorem 9.14 to show that the circumference of the circle defined by the polar equation r=ais 2蟺a.

Short Answer

Expert verified

The length of the curve is2蟺a.

Step by step solution

01

Step 1. Given Information

The function that bounds the circle isr=a.

02

Step 2. Use the formula of arc length of a polar curve

  • The region is bounded by the curve r=a.
  • The region is a circle, so the boundary arcs will be 胃=0to胃=2蟺.
  • So, the length of the curve will be calculated as follows:

鈭�02蟺(r')2+r2d胃=鈭�02蟺0+a2d胃=a鈭�02蟺d胃=a脳2蟺=2蟺a

  • Hence, the circumference of the curve is 2蟺a.

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Most popular questions from this chapter

In Exercises 24鈥�31 find all polar coordinate representations for the point given in rectangular coordinates.

6,23

In Exercises 32鈥�47 convert the equations given in polar coordinates to rectangular coordinates.

r=2cos胃

In Exercises 48鈥�55 convert the equations given in rectangular coordinates to equations in polar coordinates.

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Show that the eccentricity satisfies the equationB2=A21-e2.

Sketch the graphs of the equations

r=22+cos胃and localid="1649860998050" r=22+sin胃

What is the relationship between these graphs? What is the eccentricity of each graph?
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