Chapter 9: Q 11. (page 775)

Find the arc lengths of the curves defined by the parametric equations on the specified intervals.
$x = \sin (k t)$ , $y = \cos (k t)$ , $t \in [0, 2 π]$ , where is $k$ is constant.

Short Answer

Expert verified

The arc length is $\begin{array}{rcl} 2 k π \end{array}$ .

Step by step solution

Step 1. Given information.

The given parametric equations are $x = \sin (k t)$ and $y = \cos (k t)$ , where $k$ is a constant.

Step 2. Find the derivative of the parametric equations.

$\begin{array}{rcl} x & = & \sin (k t) \\ f' (t) & = & k \cos (k t) \end{array}$ and $\begin{array}{rcl} y & = & \cos (k t) \\ g' (t) & = & - k \sin (k t) \end{array}$

Step 3. Substitute the value of f'(t) and g'(t) in the arc length formula.

The arc length formula is $\int_{a}^{b} \sqrt{{(f' (t))}^{2} + {(g' (t))}^{2}} d t$ , where $[a, b] = [0, 2 π]$ .

$\begin{array}{rcl} \int_{a}^{b} \sqrt{{(f' (t))}^{2} + {(g' (t))}^{2}} d t & = & \int_{0}^{2 π} \sqrt{{(k \cos (k t))}^{2} + {(- k \sin (k t))}^{2}} d t \\ = & \int_{0}^{2 π} \sqrt{k^{2} \cos^{2} k t + k^{2} \sin^{2} k t} d t \\ = & k \int_{0}^{2 π} \sqrt{\cos^{2} k t + \sin^{2} k t} d t \\ = & k \int_{0}^{2 π} \sqrt{1} d t \\ = & k {[t]}_{0}^{2 π} \\ = & 2 k π \end{array}$

Step 4. Simplified answer.

Hence, the required arc length is $\begin{array}{rcl} 2 k π \end{array}$ .

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Find the arc lengths of the curves defined by the parametric equations on the specified intervals.
$x = \sin (k t)$ , $y = \cos (k t)$ , $t \in [0, 2 π]$ , where is $k$ is constant.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the derivative of the parametric equations.

Step 3. Substitute the value of f'(t) and g'(t) in the arc length formula.

Step 4. Simplified answer.

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Find the arc lengths of the curves defined by the parametric equations on the specified intervals.x=sinkt,y=coskt,t∈0,2π, where iskis constant.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the derivative of the parametric equations.

Step 3. Substitute the value of f'(t) and g'(t) in the arc length formula.

Step 4. Simplified answer.

One App. One Place for Learning.

Most popular questions from this chapter

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