# How do you find the exact length of the polar curve #r=1+sin(theta)# ?

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To find the exact length of the polar curve ( r = 1 + \sin(\theta) ), you would 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:

- ( L ) is the length of the curve.
- ( r ) is the polar function.
- ( \frac{{dr}}{{d\theta}} ) is the derivative of ( r ) with respect to ( \theta ).
- ( \alpha ) and ( \beta ) are the initial and final angles that define the portion of the curve you're interested in.

For ( r = 1 + \sin(\theta) ):

- ( r^2 = (1 + \sin(\theta))^2 ).
- ( \frac{{dr}}{{d\theta}} = \cos(\theta) ).

Substitute these values into the arc length formula and integrate over the appropriate range of ( \theta ) values to find the exact length of the polar 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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