What is the arclength of #r=-2sin(theta/4+(7pi)/8) # on #theta in [(pi)/4,(7pi)/4]#?
Arclength is given by:
Apply the Trigonometric power-reduction formula:
Integrate directly:
Insert limits:
Apply the Trigonometric sum-to-product formula:
This can probably be further simplified.
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To find the arc length of the curve defined by ( r = -2\sin\left(\frac{\theta}{4} + \frac{7\pi}{8}\right) ) on the interval ( \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] ), we can use the formula for arc length in polar coordinates:
[ L = \int_{\theta_1}^{\theta_2} \sqrt{r^2 + \left(\frac{{dr}}{{d\theta}}\right)^2} , d\theta ]
where ( r ) is the polar function and ( \frac{{dr}}{{d\theta}} ) represents its derivative with respect to ( \theta ).
By calculating the derivative of ( r ), we find ( \frac{{dr}}{{d\theta}} = -\frac{1}{2}\cos\left(\frac{\theta}{4} + \frac{7\pi}{8}\right) ).
Then, we substitute ( r ) and ( \frac{{dr}}{{d\theta}} ) into the arc length formula and integrate with respect to ( \theta ) over the given interval:
[ L = \int_{\frac{\pi}{4}}^{\frac{7\pi}{4}} \sqrt{\left(-2\sin\left(\frac{\theta}{4} + \frac{7\pi}{8}\right)\right)^2 + \left(-\frac{1}{2}\cos\left(\frac{\theta}{4} + \frac{7\pi}{8}\right)\right)^2} , d\theta ]
After simplifying, the integral becomes:
[ L = \int_{\frac{\pi}{4}}^{\frac{7\pi}{4}} \sqrt{4\sin^2\left(\frac{\theta}{4} + \frac{7\pi}{8}\right) + \frac{1}{4}\cos^2\left(\frac{\theta}{4} + \frac{7\pi}{8}\right)} , d\theta ]
This integral can be challenging to solve directly, so numerical methods or specialized software may be needed to approximate the value of the arc length.
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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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- How do you sketch the graph of the polar equation and find the tangents at the pole of #r=3sin2theta#?
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