Home > Calculus > Determining the Length of a Polar Curve

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

Answer 1

If #theta# goes from #theta_1# to #theta_2#, then the arc length of can be found by #L=int_{theta_1}^{theta_2}sqrt{theta^2+1}d theta#.

Since #r=theta#, which gives #{dr}/{d theta}=1#, we have #L=int_{theta_1}^{theta_2}sqrt{r^2+({dr}/{d theta})^2}d theta =int_{theta_1}^{theta_2}sqrt{theta^2+1}d theta#

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Answer 2

Aurora Alvarado

To find the length of the polar curve (r = \theta), where (r) is the distance from the origin to a point on the curve and (\theta) is the angle formed by the positive x-axis and the line segment connecting the origin to the point on the curve, you use the arc length formula for polar curves:

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

Substitute (r = \theta) into this formula:

[L = \int_{\alpha}^{\beta} \sqrt{\theta^2 + 1^2} , d\theta]

Evaluate this integral over the given interval to find the length of the polar curve.

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Answer from HIX Tutor

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.