Find the length of the curve {eq}\rm x = e^t cost (t), \ y = e^t sin (t),\ 0 \le t \le \pi {/eq}

Question:

Find the length of the curve {eq}\rm x = e^t cost (t), \ y = e^t sin (t),\ 0 \le t \le \pi {/eq}

Length of a curve:

The length of a curve can be defined as the distance traveled while moving from the starting point of the curve to the ending point of the curve without slipping. The length of a complex curve can be calculated using the line integral.

Answer and Explanation: 1

We have to find the length of the curve given that: {eq}\rm x = e^t cost (t), \ y = e^t sin (t),\ 0 \le t \le \pi {/eq}

We can find the length of the curve using the given formula: {eq}L = \int_{\theta_1}^{\theta_2} \sqrt{\left( \dfrac{dx}{dt} \right)^2 + \left( \dfrac{dy}{dt} \right)^2 } d\theta {/eq}

Differentiating x and y with respect to t.

{eq}\dfrac{dx}{dt} = e^t \cos (t) - e^t \sin (t) \\[0.2 cm] \dfrac{dy}{dt} = e^t \sin (t) + e^t \cos (t) {/eq}

Now, the length of the curve:

$$\begin{align} L &= \int_{\theta_1}^{\theta_2} \sqrt{\left( \dfrac{dx}{dt} \right)^2 + \left( \dfrac{dy}{dt} \right)^2 } d\theta \\[0.3 cm] &= \int_{0}^{\pi} \sqrt{\left( e^t \cos (t) - e^t \sin (t) \right)^2 + \left( e^t \sin (t) + e^t \cos (t) \right)^2 } d\theta \\[0.3 cm] &= \int_{0}^{\pi} \sqrt{e^{2t} \left( \cos (t) - \sin (t) \right)^2 + e^{2t} \left( \sin (t) + \cos (t) \right)^2 } d\theta \\[0.3 cm] &= \int_{0}^{\pi} e^{t} \sqrt{\left( \cos (t) - \sin (t) \right)^2 + \left( \sin (t) + \cos (t) \right)^2 } d\theta \\[0.3 cm] &= \int_{0}^{\pi} e^{t} \sqrt{2 \cos^2 t + 2 \sin^2 t} \\[0.3 cm] &= \int_{0}^{\pi} e^{t} \sqrt{2} \\[0.3 cm] &= \sqrt{2} \left[ e^t \right]_{0}^{\pi} \\[0.3 cm] &= \sqrt{2} \left[ e^{\pi} - e^0 \right] \\[0.3 cm] L &= \sqrt{2} \left[ e^{\pi} - 1 \right] \end{align} $$


Learn more about this topic:

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How to Find the Arc Length of a Function

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Chapter 12 / Lesson 12
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To determine the arc length of a function, integration can be used to determine the arc length over a specified interval. Learn about arc lengths and discover how to set up integrals to determine the arc length of a function over a specified interval.


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