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Integrate from the centroid of a semi circle?

Posted 1 year ago
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I am confused by this prospect. We have a semicircle with radius r whose centroid is given by 4r/3pi. Can we obtain the area of the semi circle by integrating from the centroid of the semi circle.

I saw that we could integrate from the centre of the full circle by evaluating integral through a sector like this Area of a sector

But how could we do it from the centroid of the semi circle?

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By "semicircle" you mean the arc plus a straight segment connecting the endpoints? If so, why use the region centroid when you can use the center of the circle and not change the integral shown above? That method only requires change to the angle bounds of integration.

I could integrate from the center of the circle. But my transportation problem requires me to integrate from the centroid of a semi-circle. And then centroid of a quarter circle and Generally from the centroid of a sector. I am not sure how the bounds for the general sector of a circle change to integrate from the sector centroid.

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