Respuesta :
The exact length of the polar curve in this problem is given as follows:
320,597 units.
How to calculate the length of a polar curve?
The definition of a polar curve is given as follows:
r(θ).
Then the length of the curve over an interval a ≤ θ ≤ b is given by the definite integral presented as follows:
[tex]L = \int_{a}^{b} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} d\theta[/tex]
The curve in this problem is given as follows:
[tex]r = e^{2\theta}[/tex]
The derivative is given as follows:
[tex]\frac{dr}{d\theta} = 2e^{2\theta}[/tex]
Then the arc length is given by the integral presented as follows:
[tex]L = \int_{0}^{2\pi} \sqrt{e^{4\theta} + 4e^{4\theta}} d\theta[/tex]
[tex]L = \int_{0}^{2\pi} \sqrt{5e^{4\theta}} d\theta[/tex]
[tex]L = \sqrt{5}(\int_{0}^{2\pi} e^{2\theta} d\theta)[/tex]
[tex]L = \frac{\sqrt{5}}{2}(e^{2\theta})|_{\theta = 0}^{\theta = 2\pi}[/tex]
Applying the Fundamental Theorem of Calculus to solve the double integral, the length of the arc is givena s follows:
[tex]L = \frac{\sqrt{5}}{2}(e^{4\pi} - 1)[/tex]
L = 320,597 units.
More can be learned about the length of a polar curve at https://brainly.com/question/16415788
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