Chapter 9: Q. 47 (page 756)

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

Short Answer

Expert verified

The length of the curve is $\begin{array}{rcl} 2 π a \end{array}$ .

Step by step solution

Step 1. Given Information

The function that bounds the circle is $r = a$ .

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 $θ = 0 to θ = 2 π$ .
So, the length of the curve will be calculated as follows:

$\begin{array}{rcl} \int_{0}^{2 π} \sqrt{{(r')}^{2} + r^{2}} d θ & = & \int_{0}^{2 π} \sqrt{0 + a^{2}} d θ \\ = & a \int_{0}^{2 π} d θ \\ = & a \times 2 π \\ = & 2 π a \end{array}$

Hence, the circumference of the curve is $\begin{array}{rcl} 2 π a \end{array}$ .

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

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