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Chapter 11: Problem 31

Find the length of the following polar curves. The complete circle \(r=a \sin \theta,\) where \(a>0\)

Short Answer

Expert verified
The arc length of the complete circle is \(2\pi a\).

Step by step solution

01

Determine the bounds for theta

As we have to find the length of the complete polar curve, we should determine the range of \(\theta\) for which the curve makes a complete circle. Since this polar curve represents a circle, the limits for \(\theta\) are from 0 to \(2\pi\).
02

Find the derivative of r with respect to theta

We differentiate the polar equation \(r = a \sin \theta\) with respect to \(\theta\): \(\frac{dr}{d\theta} = a \cos \theta\)
03

Use the arc length formula

Now, we apply the formula for arc length of a polar curve: \(L = \int_{\alpha}^{\beta} \sqrt{r^2 + (\frac{dr}{d\theta})^2} d\theta\) Plug in the values we found: \(L = \int_{0}^{2\pi} \sqrt{(a\sin\theta)^2 + (a\cos\theta)^2} d\theta\)
04

Simplify the expression inside the square root

Simplify the expression inside the square root: \((a\sin\theta)^2 + (a\cos\theta)^2 = a^2\sin^2\theta + a^2\cos^2\theta = a^2(\sin^2\theta+\cos^2\theta)=a^2\) So, \(L=\int_{0}^{2\pi} \sqrt{a^2} d\theta\)
05

Calculate the integral

Now, we calculate the integral to get the arc length of the polar curve: \(L = \int_{0}^{2\pi} a d\theta = a(\theta\Big|_{0}^{2\pi})\) \(L= a(2\pi - 0)\) \(L = 2\pi a\) So, the length of the complete circle for the given polar curve is \(2\pi a\).

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Evaluate the following definite integrals. $$\int_{0}^{\ln 2}\left(e^{-t} \mathbf{i}+2 e^{2 t} \mathbf{j}-4 e^{t} \mathbf{k}\right) d t$$

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