Prove that the arc length of the polar curve #r = a(1-cos theta)# is #8a#?
The polar arc length of a curve is given by:
We have:
Thus:
So, the arc length is:
Using the trig identity:
We then have:
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To find the arc length of the polar curve ( r = a(1 - \cos(\theta)) ), you can use the formula for arc length in polar coordinates:
[ L = \int_{\alpha}^{\beta} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} , d\theta ]
First, find ( \frac{dr}{d\theta} ):
[ \frac{dr}{d\theta} = \frac{d}{d\theta}(a(1 - \cos(\theta))) ]
[ = a \cdot \frac{d}{d\theta}(1 - \cos(\theta)) ]
[ = a \cdot \sin(\theta) ]
Now, substitute ( r = a(1 - \cos(\theta)) ) and ( \frac{dr}{d\theta} = a\sin(\theta) ) into the formula for arc length:
[ L = \int_{0}^{2\pi} \sqrt{(a(1 - \cos(\theta)))^2 + (a\sin(\theta))^2} , d\theta ]
[ = \int_{0}^{2\pi} \sqrt{a^2(1 - 2\cos(\theta) + \cos^2(\theta)) + a^2\sin^2(\theta)} , d\theta ]
[ = \int_{0}^{2\pi} \sqrt{a^2 - 2a^2\cos(\theta) + a^2\cos^2(\theta) + a^2\sin^2(\theta)} , d\theta ]
[ = \int_{0}^{2\pi} \sqrt{a^2 - 2a^2\cos(\theta) + a^2} , d\theta ]
[ = \int_{0}^{2\pi} \sqrt{2a^2(1 - \cos(\theta))} , d\theta ]
[ = \int_{0}^{2\pi} \sqrt{2}a\sqrt{1 - \cos(\theta)} , d\theta ]
[ = \sqrt{2}a \int_{0}^{2\pi} \sqrt{1 - \cos(\theta)} , d\theta ]
Since ( \sqrt{1 - \cos(\theta)} ) is an even function, the integral over one period is twice the integral from 0 to ( \pi ).
[ = 2\sqrt{2}a \int_{0}^{\pi} \sqrt{1 - \cos(\theta)} , d\theta ]
[ = 2\sqrt{2}a \cdot \frac{\pi}{2} ]
[ = \sqrt{2}\pi a ]
Given that ( a ) is a positive constant, ( \sqrt{2}\pi a = 8a ). Therefore, the arc length of the polar curve ( r = a(1 - \cos(\theta)) ) is indeed ( 8a ).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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- What is the area enclosed by #r=theta^2-2sintheta # for #theta in [pi/4,pi]#?
- What is the slope of the tangent line of #r=2theta^2-3thetacos(2theta-(pi)/3)# at #theta=(-5pi)/3#?
- What is the equation of the tangent line of #r=cos(theta-pi/4) +sin^2(theta+pi)-theta# at #theta=(-13pi)/4#?
- What is the distance between the following polar coordinates?: # (3,(5pi)/4), (1,(pi)/8) #

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