Using integration, find the area of the region bounded by (2, 5)

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 Multiple Choice QuestionsShort Answer Type

41.

Find the area bounded by the curve y = sin x between x = 0 and x = 2 straight pi.

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 Multiple Choice QuestionsLong Answer Type

42.

Using integration, find the area of the triangular region whose sides have the equations y = 2 x + 1, y = 3 x + 1 and x = 4.

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43. Using the method of integration find the area of the region bounded by lines:
2 x + y = 4, 3 x - 2 y = 6 and x - 3 y + 5 = 0.
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44.

Using integration, find the area of the region bounded by (2, 5), (4, 7) and (6, 2).


Let A(2, 5), B(4, 7), C(6, 2) be vertices of ∆ABC. from A, B, C draw AL. BM. CN ⊥ x-axis.
The equation of AB is
                            straight y minus 5 space equals space fraction numerator 7 minus 5 over denominator 4 minus 2 end fraction left parenthesis straight x minus 2 right parenthesis
or                    straight y minus 5 space equals 2 over 2 left parenthesis straight x minus 2 right parenthesis
or                       straight y minus 5 space equals space straight x minus 2
or                          straight y space equals space straight x plus 3
The equation of BC is
                   straight y minus 7 space equals space fraction numerator 2 minus 7 over denominator 6 minus 4 end fraction left parenthesis straight x minus 4 right parenthesis space space space space or space space space straight y minus 7 space equals space minus 5 over 2 left parenthesis straight x minus 4 right parenthesis
or                 straight y minus 7 space equals space minus fraction numerator 5 straight x over denominator 2 end fraction plus 10 space space space space or space space straight y equals negative fraction numerator 5 straight x over denominator 2 end fraction plus 17


The equation of CA is
                       straight y minus 2 space equals space fraction numerator 5 minus 2 over denominator 2 minus 6 end fraction left parenthesis straight x minus 6 right parenthesis space space space or space straight y minus 2 space equals space minus 3 over 4 left parenthesis straight x minus 6 right parenthesis
or                     straight y minus 2 space equals space minus fraction numerator 3 straight x over denominator 4 end fraction plus 9 over 2 space space space space space space or space space space space straight y space equals negative fraction numerator 3 straight x over denominator 4 end fraction plus 13 over 2
Area of increment ABC =  Area ALMB+ area BMNC - area ALNC
                      equals space integral subscript 2 superscript 4 left parenthesis straight x plus 3 right parenthesis space dx plus integral subscript 4 superscript 6 open parentheses negative fraction numerator 5 straight x over denominator 2 end fraction plus 17 close parentheses space dx space minus space integral subscript 2 superscript 6 open parentheses negative fraction numerator 3 straight x over denominator 4 end fraction plus 13 over 2 close parentheses dx
         equals space open square brackets straight x squared over 2 plus 3 straight x close square brackets subscript 2 superscript 4 plus open square brackets fraction numerator negative 5 straight x squared over denominator 4 end fraction plus 17 straight x close square brackets subscript 4 superscript 6 space minus space open square brackets negative fraction numerator 3 straight x squared over denominator 8 end fraction plus fraction numerator 13 space straight x over denominator 2 end fraction close square brackets subscript 2 superscript 6
     equals left parenthesis 8 plus 12 right parenthesis minus left parenthesis 2 plus 6 right parenthesis plus left parenthesis negative 45 plus 102 right parenthesis minus left parenthesis negative 20 plus 68 right parenthesis minus open parentheses negative 27 over 2 plus 39 close parentheses plus open parentheses negative 3 over 2 plus 13 close parentheses
equals 20 minus 8 plus 57 minus 48 minus 51 over 2 plus 23 over 2
equals fraction numerator 40 minus 16 plus 114 minus 96 minus 51 plus 23 over denominator 2 end fraction equals 14 over 2 space equals space 7 space sq. space units. space

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45. Using the method of integration, find the area of the triangle ABC, co-ordinates of whose vertices are A (2, 0), B (4, 5), C (6, 3).
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46.

Using integration, find the area of the region bounded by the triangle whose vertices are (-1, 1), (0, 5) and (3, 2).

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47. Using integration, find the area of the triangle ABC whose vertices have coordinates A (3, 0), B(4, 6) and C (6, 2).
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48. Using integration, find the area of the triangle ABC whose vertices are A (3, 0) B (4, 5) and C (5, 1).
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 Multiple Choice QuestionsShort Answer Type

49.

Using integration, find the area of the region bounded by the triangle whose vertices are (1, 0), (2, 2) and (3, 1).

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 Multiple Choice QuestionsLong Answer Type

50.

Using integration find the area of region bounded by the triangle whose vertices are (-1, 0), (1, 3) and (3, 2).

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