Chapter 9: Q. 48 (page 756)

Use Theorem 9.14 to show that the circumference of the circle defined by the polar equation $r = 2 a \sin θ$ is $2 π a$ .

Short Answer

Expert verified

The circumference is found to be $2 π a$ by using the formula of length of polar curves.

Step by step solution

Step 1. Given Information

The function that bounds the circle is $r = 2 a \sin θ$ .

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

The region is bounded by the curve $r = 2 a \sin θ$ .
The region is a circle, so the boundary arcs will be $θ = 0 to θ = 2 π$ .
However, the function traces itself twice in this domain.
So, the length of the curve will be calculated as follows:

role="math" localid="1650343133331" $\frac{1}{2} \begin{array}{rcl} \int \end{array} \begin{array}{rcl} 0 2 π \end{array} \begin{array}{rcl} \sqrt{{(r')}^{2} + r^{2}} \end{array} \begin{array}{rcl} d \end{array} \begin{array}{rcl} θ \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} \frac{1}{2} \end{array} \begin{array}{rcl} \int \end{array} \begin{array}{rcl} 0 2 π \end{array} \begin{array}{rcl} \sqrt{{(2 a \cos θ)}^{2} + {(2 a \sin θ)}^{2}} \end{array} \begin{array}{rcl} d \end{array} \begin{array}{rcl} θ \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} \frac{1}{2} \end{array} \begin{array}{rcl} \times \end{array} \begin{array}{rcl} 2 \end{array} \begin{array}{rcl} a \end{array} \begin{array}{rcl} \int \end{array} \begin{array}{rcl} 0 2 π \end{array} \begin{array}{rcl} \sqrt{1} \end{array} \begin{array}{rcl} d \end{array} \begin{array}{rcl} θ \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} a \end{array} \begin{array}{rcl} \times \end{array} \begin{array}{rcl} 2 \end{array} \begin{array}{rcl} π \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} 2 \end{array} \begin{array}{rcl} π \end{array} \begin{array}{rcl} a \end{array}$

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Use Theorem 9.14 to show that the circumference of the circle defined by the polar equation $r = 2 a \sin θ$ is $2 π a$ .

Short Answer

Step by step solution

Step 1. Given Information

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

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Use Theorem 9.14 to show that the circumference of the circle defined by the polar equation r=2asinθis 2πa.

Short Answer

Step by step solution

Step 1. Given Information

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Discrete Mathematics

Probability and Statistics

Mechanics Maths

Geometry

Logic and Functions

Study anywhere. Anytime. Across all devices.

Use Theorem 9.14 to show that the circumference of the circle defined by the polar equation $r = 2 a \sin θ$ is $2 π a$ .