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bigevil

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## Homework Statement

Here, I have two ways of finding the y-coordinate centroid of a semicircular arc using polar coordinates.

First one is considering a circle of radius a, centred at the origin. What I have done is [tex]\int ds = \int_0^{\pi} a d\theta = \pi a[/tex] and then [tex]\int y ds = \int r^2 sin \theta d\theta = a^2 \int_0^{\pi} sin \theta d\theta = 2a^2[/tex], then y-centroid is 2a/π. I'm quite sure this is the right answer.

Second is a more roundabout method. It uses a circle with equation [tex]r=2a cos \theta[/tex]. A right angle triangle fits inside the circle and the angle between the side passing through the origin and the x-axis is theta. But I do not get a similar answer. Here my method is to use [tex]\int y ds = \int r sin\theta \sqrt{r^2 + (\frac{dr}{d\theta})^2} d\theta = \int_0^{\frac{\pi}{2}} 2a cos \theta (2a) d\theta = 4a^2[/tex], where the limits of the integral run from 0 to half pi due to how the angle is set up. (From my working, the messy square roots part becomes just 2a.) The y-centroid for the semicircle is 4a/π, which is wrong. Can someone advise me where I have gone wrong?

The y-coordinate centroid should be the same for both as the difference between the two figures is that one is shifted a-units along the x-axis.

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