Expression 16: "I" Subscript, "i" "n" "t" "e" "r" "e" "s" "t" , Baseline equals "I" left bracket, left brace, "P" . "x" minus "R" less than "x" Subscript, "m" "a" "x" , Baseline , right brace left brace, "P" . "x" plus "R" greater than "x" Subscript, "m" "i" "n" , Baseline , right brace equals 1 , right bracketIinterest=IP.x−R<xmaxP.x+R>xmin=1
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Expression 17: "y" Subscript, "m" "i" "d" "R" "a" "w" "B" "o" "t" "t" "o" "m" , Baseline equals left parenthesis, "P" . "y" minus StartRoot, "R" squared minus left parenthesis, "P" . "x" minus "x" Subscript, "m" "i" "d" , Baseline , right parenthesis squared , EndRoot , right parenthesis left bracket, "I" Subscript, "i" "n" "t" "e" "r" "e" "s" "t" , Baseline , right bracketymidRawBottom=P.y−R2−P.x−xmid2Iinterest
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Expression 18: "y" Subscript, "m" "i" "d" "R" "a" "w" "T" "o" "p" , Baseline equals left parenthesis, "P" . "y" plus StartRoot, "R" squared minus left parenthesis, "P" . "x" minus "x" Subscript, "m" "i" "d" , Baseline , right parenthesis squared , EndRoot , right parenthesis left bracket, "I" Subscript, "i" "n" "t" "e" "r" "e" "s" "t" , Baseline , right bracketymidRawTop=P.y+R2−P.x−xmid2Iinterest
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Expression 19: "I" Subscript, "t" "o" "p" "R" "a" "w" , Baseline equals "I" Subscript, "i" "n" "t" "e" "r" "e" "s" "t" , Baseline left bracket, 1 equals left bracket, min left parenthesis, left bracket, left brace, "i" equals "j" , "y" Subscript, "m" "i" "d" "R" "a" "w" "B" "o" "t" "t" "o" "m" , Baseline left bracket, "j" , right bracket greater than "y" Subscript, "m" "i" "d" "R" "a" "w" "T" "o" "p" , Baseline left bracket, "i" , right bracket , "y" Subscript, "m" "i" "d" "R" "a" "w" "T" "o" "p" , Baseline left bracket, "j" , right bracket less than "y" Subscript, "m" "i" "d" "R" "a" "w" "T" "o" "p" , Baseline left bracket, "i" , right bracket , right brace for "j" equals left bracket, 1... "I" Subscript, "i" "n" "t" "e" "r" "e" "s" "t" , Baseline . , right bracket , right bracket , right parenthesis for "i" equals left bracket, 1... "I" Subscript, "i" "n" "t" "e" "r" "e" "s" "t" , Baseline . , right bracket , right bracket , right bracketItopRaw=Iinterest1=mini=j,ymidRawBottomj>ymidRawTopi,ymidRawTopj<ymidRawTopiforj=1...Iinterest.lengthfori=1...Iinterest.length
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Expression 20: "I" Subscript, "b" "o" "t" "t" "o" "m" "R" "a" "w" , Baseline equals "I" Subscript, "i" "n" "t" "e" "r" "e" "s" "t" , Baseline left bracket, 1 equals left bracket, min left parenthesis, left bracket, left brace, "i" equals "j" , "y" Subscript, "m" "i" "d" "R" "a" "w" "B" "o" "t" "t" "o" "m" , Baseline left bracket, "j" , right bracket greater than "y" Subscript, "m" "i" "d" "R" "a" "w" "B" "o" "t" "t" "o" "m" , Baseline left bracket, "i" , right bracket , "y" Subscript, "m" "i" "d" "R" "a" "w" "T" "o" "p" , Baseline left bracket, "j" , right bracket less than "y" Subscript, "m" "i" "d" "R" "a" "w" "B" "o" "t" "t" "o" "m" , Baseline left bracket, "i" , right bracket , right brace for "j" equals left bracket, 1... "I" Subscript, "i" "n" "t" "e" "r" "e" "s" "t" , Baseline . , right bracket , right bracket , right parenthesis for "i" equals left bracket, 1... "I" Subscript, "i" "n" "t" "e" "r" "e" "s" "t" , Baseline . , right bracket , right bracket , right bracketIbottomRaw=Iinterest1=mini=j,ymidRawBottomj>ymidRawBottomi,ymidRawTopj<ymidRawBottomiforj=1...Iinterest.lengthfori=1...Iinterest.length
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Expression 21: "I" Subscript, "t" "o" "p" , Baseline equals sort left parenthesis, "I" Subscript, "t" "o" "p" "R" "a" "w" , Baseline , "y" Subscript, "m" "i" "d" "R" "a" "w" "T" "o" "p" , Baseline left bracket, "I" Subscript, "t" "o" "p" "R" "a" "w" , Baseline , right bracket , right parenthesisItop=sortItopRaw,ymidRawTopItopRaw
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Expression 22: "I" Subscript, "b" "o" "t" "t" "o" "m" , Baseline equals sort left parenthesis, "I" Subscript, "b" "o" "t" "t" "o" "m" "R" "a" "w" , Baseline , "y" Subscript, "m" "i" "d" "R" "a" "w" "B" "o" "t" "t" "o" "m" , Baseline left bracket, "I" Subscript, "b" "o" "t" "t" "o" "m" "R" "a" "w" , Baseline , right bracket , right parenthesisIbottom=sortIbottomRaw,ymidRawBottomIbottomRaw
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Expression 23: "A" Subscript, "r" "e" "a" "S" "l" "i" "c" "e" , Baseline left parenthesis, "t" , right parenthesis equals total left parenthesis, "A" Subscript, "r" "e" "a" "B" "e" "t" "w" "e" "e" "n" "A" "r" "c" "s" , Baseline left parenthesis, "P" left bracket, "I" Subscript, "t" "o" "p" , Baseline , right bracket , "R" left bracket, "I" Subscript, "t" "o" "p" , Baseline , right bracket , "P" left bracket, "I" Subscript, "b" "o" "t" "t" "o" "m" , Baseline , right bracket , "R" left bracket, "I" Subscript, "b" "o" "t" "t" "o" "m" , Baseline , right bracket , right parenthesis , right parenthesisAreaSlicet=totalAreaBetweenArcsPItop,RItop,PIbottom,RIbottom
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Area between arcs
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Area from x_min to y_min, of the section bounded: on left by x=x_min; on right by x=x_max; on top by an arc of the circle with center P1 and radius R1; on bottom by an arc of the circle with center P2 and radius R2.
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Expression 26: . "A" Subscript, "r" "e" "a" "B" "e" "t" "w" "e" "e" "n" "A" "r" "c" "s" , Baseline left parenthesis, "P" Subscript, 1 , Baseline , "R" Subscript, 1 , Baseline , "P" Subscript, 2 , Baseline , "R" Subscript, 2 , Baseline , right parenthesis equals Start integral from "x" Subscript, "m" "i" "n" , Baseline to "x" Subscript, "m" "a" "x" , Baseline , end integral, left parenthesis, left parenthesis, "P" Subscript, 1 , Baseline . "y" plus StartRoot, "R" squared minus left parenthesis, "x" minus "P" Subscript, 1 , Baseline . "x" , right parenthesis squared , EndRoot , right parenthesis minus left parenthesis, "P" Subscript, 2 , Baseline . "y" minus StartRoot, "R" squared minus left parenthesis, "x" minus "P" Subscript, 2 , Baseline . "x" , right parenthesis squared , EndRoot , right parenthesis , right parenthesis "d" "x".AreaBetweenArcsP1,R1,P2,R2=∫xmaxxminP1.y+R21−x−P1.x2−P2.y−R22−x−P2.x2dx
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Expression 27: . "A" Subscript, "r" "e" "a" "B" "e" "t" "w" "e" "e" "n" "A" "r" "c" "s" , Baseline left parenthesis, "P" Subscript, 1 , Baseline , "R" Subscript, 1 , Baseline , "P" Subscript, 2 , Baseline , "R" Subscript, 2 , Baseline , right parenthesis equals left parenthesis, "x" Subscript, "m" "a" "x" , Baseline minus "x" Subscript, "m" "i" "n" , Baseline , right parenthesis left parenthesis, "P" Subscript, 1 , Baseline . "y" minus "P" Subscript, 2 , Baseline . "y" , right parenthesis plus Start integral from "x" Subscript, "m" "i" "n" , Baseline to "x" Subscript, "m" "a" "x" , Baseline , end integral, left parenthesis, StartRoot, "R" squared minus left parenthesis, "x" minus "P" Subscript, 1 , Baseline . "x" , right parenthesis squared , EndRoot plus StartRoot, "R" squared minus left parenthesis, "x" minus "P" Subscript, 2 , Baseline . "x" , right parenthesis squared , EndRoot , right parenthesis "d" "x".AreaBetweenArcsP1,R1,P2,R2=xmax−xminP1.y−P2.y+∫xmaxxminR21−x−P1.x2+R22−x−P2.x2dx
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indefinite integral of sqrt(R^2-x^2)
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Expression 29: "f" Subscript, "i" "n" "d" "e" "f" , Baseline left parenthesis, "x" , "r" , right parenthesis equals 1 half "x" StartRoot, "r" squared minus "x" squared , EndRoot plus 1 half "r" squared arc tangent left parenthesis, StartFraction, "x" Over StartRoot, "r" squared minus "x" squared , EndRoot , EndFraction , right parenthesisfindefx,r=12xr2−x2+12r2arctanxr2−x2
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Expression 30: "f" Subscript, "d" "e" "f" , Baseline left parenthesis, "x" Subscript, 1 , Baseline , "x" Subscript, 2 , Baseline , "r" , right parenthesis equals "f" Subscript, "i" "n" "d" "e" "f" , Baseline left parenthesis, "x" Subscript, 2 , Baseline , "r" , right parenthesis minus "f" Subscript, "i" "n" "d" "e" "f" , Baseline left parenthesis, "x" Subscript, 1 , Baseline , "r" , right parenthesisfdefx1,x2,r=findefx2,r−findefx1,r
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Expression 31: "A" Subscript, "r" "e" "a" "B" "e" "t" "w" "e" "e" "n" "A" "r" "c" "s" , Baseline left parenthesis, "P" Subscript, 1 , Baseline , "R" Subscript, 1 , Baseline , "P" Subscript, 2 , Baseline , "R" Subscript, 2 , Baseline , right parenthesis equals left parenthesis, "x" Subscript, "m" "a" "x" , Baseline minus "x" Subscript, "m" "i" "n" , Baseline , right parenthesis left parenthesis, "P" Subscript, 1 , Baseline . "y" minus "P" Subscript, 2 , Baseline . "y" , right parenthesis plus "f" Subscript, "d" "e" "f" , Baseline left parenthesis, "x" Subscript, "m" "i" "n" , Baseline minus "P" Subscript, 1 , Baseline . "x" , "x" Subscript, "m" "a" "x" , Baseline minus "P" Subscript, 1 , Baseline . "x" , "R" Subscript, 1 , Baseline , right parenthesis plus "f" Subscript, "d" "e" "f" , Baseline left parenthesis, "x" Subscript, "m" "i" "n" , Baseline minus "P" Subscript, 2 , Baseline . "x" , "x" Subscript, "m" "a" "x" , Baseline minus "P" Subscript, 2 , Baseline . "x" , "R" Subscript, 2 , Baseline , right parenthesisAreaBetweenArcsP1,R1,P2,R2=xmax−xminP1.y−P2.y+fdefxmin−P1.x,xmax−P1.x,R1+fdefxmin−P2.x,xmax−P2.x,R2