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February 4th, 2015, 07:28 AM   #1
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Area of shape formed by circles

Please help! This problem should be pretty interesting to solve.

The problem is in the attached picture. Many thanks!
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 February 4th, 2015, 08:15 AM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,560 Thanks: 2561 Math Focus: Mainly analysis and algebra It's easy if you use the graph (hint: don't use complete circles, use the square formed by the centres of the circles and their intersections).
 February 4th, 2015, 01:49 PM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,560 Thanks: 2561 Math Focus: Mainly analysis and algebra Alternatively (using some calculus), note that the right hand circle has equation $(x-2)^2 + y^2 = 4$. It has a segment cut off by the line $y = x$. The distance from this line to the centre of the circle is $\sqrt2$. Thus the area, $A$, of the whole shape is given by$$A = 4 \int_{2-\sqrt2}^4 \sqrt{4 - (x-2)^2}\,\mathrm d x$$ To get this, we have effectively rotated the circle by $\frac\pi4$ so that the chord is vertical. Since the shape is symmetrical about this chord and the perpendicular to the chord which passes through the centre of the two circles, the given integral emerges.

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