Find the area of the region enclosed between the two circles  

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 Multiple Choice QuestionsLong Answer Type

51. Find the area of the region enclosed between the two circles x2 + y2 = 4 and (x - 2)2 + y2 = 4.
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52. Find the area of the region enclosed between the two circles  x2 + y2 = 1 and (x - 1)2 + y2 = 1. 


The equation of two circles are
                        straight x squared plus straight y squared space equals space 1                          ...(1)
   and         left parenthesis straight x minus 1 right parenthesis squared plus straight y squared space equals space 1                       ...(2)
Centre of circle (1) is O (0, 0) and radius OA = 1
Centre of circle (2) is A (1, 0) and radius AO = 1
Subtracting (1) from (2), we get,
negative 2 straight x plus 1 space equals space 0 space space space space space space rightwards double arrow space space space space straight x space equals space 1 half
Putting straight x space equals space 1 half in (1), we get,
                 1 fourth plus straight y squared space equals space 1 space space space space space space rightwards double arrow space space space space straight y squared equals space 3 over 4
rightwards double arrow space space space space space straight y space equals space plus-or-minus fraction numerator square root of 3 over denominator 2 end fraction



therefore     points of intersection of circles (1) and (2) are
                  straight P open parentheses 1 half comma space fraction numerator square root of 3 over denominator 2 end fraction close parentheses space space and space straight Q space open parentheses 1 half comma space minus fraction numerator square root of 3 over denominator 2 end fraction close parentheses
        Required area =  Area of region OQAP
                      = 2 (area of region OMAP)
                     = 2 (area of region OMPO + area  of region MAPM)
                      equals space 2 open square brackets integral subscript 0 superscript 1 divided by 2 end superscript square root of 1 minus left parenthesis straight x minus 1 right parenthesis squared end root dx plus integral subscript 1 divided by 2 end subscript superscript 1 square root of 1 minus straight x squared end root close square brackets
                       equals space 2 open square brackets fraction numerator left parenthesis straight x minus 1 right parenthesis thin space square root of 1 minus left parenthesis straight x minus 1 right parenthesis squared end root over denominator 2 end fraction plus 1 half sin to the power of negative 1 end exponent left parenthesis straight x minus 1 right parenthesis close square brackets subscript 0 superscript 1 half end superscript plus 2 open square brackets fraction numerator straight x square root of 1 minus straight x squared end root over denominator 2 end fraction plus 1 half sin to the power of negative 1 end exponent straight x close square brackets subscript 1 half end subscript superscript 1
                       equals space 2 open square brackets open curly brackets fraction numerator open parentheses begin display style 1 half end style minus 1 close parentheses square root of 1 minus open parentheses begin display style 1 half end style minus 1 close parentheses squared end root over denominator 2 end fraction plus 1 half sin to the power of negative 1 end exponent open parentheses 1 half minus 1 close parentheses close curly brackets close square brackets
negative open curly brackets fraction numerator left parenthesis 0 minus 1 right parenthesis space square root of 1 minus left parenthesis 0 minus 1 right parenthesis squared end root over denominator 2 end fraction plus 1 half sin to the power of negative 1 end exponent left parenthesis 0 minus 1 right parenthesis close curly brackets plus 2 space open square brackets open parentheses 0 plus 1 half sin to the power of negative 1 end exponent 1 close parentheses minus fraction numerator begin display style 1 half end style square root of 1 minus begin display style 1 fourth end style end root over denominator 2 end fraction plus 1 half sin to the power of negative 1 end exponent open parentheses 1 half close parentheses close square brackets
               equals space 2 open square brackets fraction numerator negative begin display style 1 half end style cross times begin display style fraction numerator square root of 3 over denominator 2 end fraction end style over denominator 2 end fraction plus 1 half sin to the power of negative 1 end exponent open parentheses negative 1 half close parentheses minus 0 minus 1 half sin to the power of negative 1 end exponent left parenthesis negative 1 right parenthesis close square brackets
                                                                                                 plus space 2 space open square brackets 1 half sin to the power of negative 1 end exponent 1 minus fraction numerator begin display style 1 half end style cross times begin display style fraction numerator square root of 3 over denominator 2 end fraction end style over denominator 2 end fraction minus 1 half sin to the power of negative 1 end exponent 1 half close square brackets

    equals space minus fraction numerator square root of 3 over denominator 4 end fraction plus open parentheses negative straight pi over 6 close parentheses minus open parentheses negative straight pi over 2 close parentheses plus straight pi over 2 minus fraction numerator square root of 3 over denominator 4 end fraction minus straight pi over 6
equals space open parentheses negative straight pi over 6 plus straight pi over 2 plus straight pi over 2 minus straight pi over 6 close parentheses minus 2 cross times fraction numerator square root of 3 over denominator 4 end fraction space equals space open parentheses 2 straight pi over 3 minus fraction numerator square root of 3 over denominator 2 end fraction close parentheses space sq. space units.
                   

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53. Find the area of the region enclosed between the two circles x2 + y2 = 1 and open parentheses straight x minus 1 half close parentheses squared plus straight y squared space equals space 1.
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 Multiple Choice QuestionsShort Answer Type

54. Find the area of the region bounded by two parabola 4 y = x2 and 4 x = y2
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 Multiple Choice QuestionsLong Answer Type

55. Find the area included between the two curves y2 = 9x and ,x2 = 9y. Also draw the rough sketch.
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 Multiple Choice QuestionsShort Answer Type

56.

Calculate the area of the region bounded by the y = x2 and x = y2.

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 Multiple Choice QuestionsLong Answer Type

57. Draw a graph of y2 = 16 x and x2 = 16y, and evaluate the area between them.
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58. Find the area of the region included between the parabolas y2 = 4 a x and x2 = 4 a y, a > 0.    
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59. Draw the rough sketch of y2 = x + 1 and y2 = - x + 1 and determine the area enclosed by the two curves.
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60. Find the area of the region
{(x, y): 0 ≤ y ≤ x2 + 1 , 0 ≤ y ≤ x + 1, 0 ≤ x ≤ 2}.
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