Calculus Calculus Math Forum

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

The problem is in the attached picture. Many thanks!
Attached Images 10961816_10153092519289700_1936058838_n.jpg (31.1 KB, 7 views) February 4th, 2015, 08:15 AM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,697 Thanks: 2681 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: 7,697 Thanks: 2681 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. Tags area, circles, formed, shape Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Luckman1 Calculus 2 May 21st, 2014 02:53 PM nubshat Calculus 2 November 20th, 2013 08:08 AM mathismydoom Algebra 6 September 29th, 2012 06:34 AM tiba Algebra 3 January 15th, 2012 07:58 PM NASAorbust Algebra 15 March 21st, 2011 09:50 AM

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