# How do you find the length of the polar curve #r=5^theta# ?

You can find the length of this polar curve by applying the formula for Arc Length for Parametric Equations:

Giving us an answer of:

Process:

Plugging this into our formula, we have:

Distribute the exponent:

is actually a constant, which means it can be pulled out of the integral entirely:

We now have:

Simplifying, we arrive at our final answer:

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To find the length of the polar curve ( r = 5^\theta ), you can use the arc length formula for polar curves:

[ L = \int_{\alpha}^{\beta} \sqrt{r^2 + \left(\frac{{dr}}{{d\theta}}\right)^2} , d\theta ]

where ( r = 5^\theta ).

First, find ( \frac{{dr}}{{d\theta}} ):

[ \frac{{dr}}{{d\theta}} = \frac{{d}}{{d\theta}}(5^\theta) = \ln(5) \cdot 5^\theta ]

Now, substitute ( r ) and ( \frac{{dr}}{{d\theta}} ) into the arc length formula and integrate it over the appropriate range of ( \theta ) values to find the length of the curve.

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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.

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