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June 13th, 2013, 04:27 PM   #1
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Polar function from a (circle and square) shape

http://i.imgur.com/sUM8fwH.gif

I'm trying to find the area of the blue shape (technically i'm trying to find the centroid). How would i setup the integral in polar form?

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June 13th, 2013, 06:50 PM   #2
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Re: Polar function from a (circle and square) shape

If the image is accurate, so that the the axes are at a tangent to the circle, then an integral is not really necessary. We have a quarter circle and a square.





Hence the area shown is



If you want to do it with integration, just do the same thing. Consider the graph illustrating the two functions:





Then, being explicit with your request, we can find the area of the bottom-left quadrant of the circle described by :



And the square is just the area bound between the line y(x) and the x-axis from -R to 0. Notice that since the line is below the axis, we're going to get the negative area. To correct, we'll just multiply by -1:



And then the area of the blue is the square minus the quarter circle:



To find the centroid of an object with upper bound f(x) and lower bound g(x) over the region , the following formulae can be applied to find the centroid :





Where A is the area of the shape enclosed by f and g. Formulae thanks to wiki. If we let:



And



Then it's clear the area enclosed is , since f and g are two halves of a circle with radius R. We will look at the region -R<x<R. Then the centroid co-ordinates are:



The substitution is elementary, but sufficed to say, . Also, we have:



And it is shown the centroid is .
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June 14th, 2013, 02:00 PM   #3
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Re: Polar function from a (circle and square) shape

thank you very much
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