Using integration find the area of region bounded by the triang

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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).

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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

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50.

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


Let A(-1, 0), B(1, 3), C(3, 2) be the vertices of the given triangle. 
The equation of AB is
                   straight y minus 0 space equals fraction numerator 3 minus 0 over denominator 1 plus 1 end fraction left parenthesis straight x plus 1 right parenthesis
or                 straight y equals space 3 over 2 left parenthesis straight x plus 1 right parenthesis              ...(1)

The equation of BC is
                    straight y minus 3 space equals space fraction numerator 2 minus 3 over denominator 3 minus 1 end fraction left parenthesis straight x minus 1 right parenthesis
or                straight y minus 3 space equals negative 1 half left parenthesis straight x minus 1 right parenthesis space or space straight y minus 3 space equals space minus 1 half straight x plus 1 half
or                  straight y equals negative 1 half straight x plus 7 over 2                        ...(2)
The equation of CA is
                  straight y minus 2 space equals space fraction numerator 0 minus 2 over denominator negative 1 minus 3 end fraction left parenthesis straight x minus 3 right parenthesis space space or space space straight y space minus space 2 space equals space 1 half left parenthesis straight x minus 3 right parenthesis
or         straight y minus 2 space equals space 1 half straight x minus 3 over 2 space space space space or space space space space straight y space equals space 1 half straight x plus 1 half         ...(3)
From B, draw BM ⊥ x-axis and from C, draw CN ⊥ x-axis.
Required area = Area of ∆ABC
= Area of increment AMB + area of region BMNC - area of increment ANC
equals space integral subscript negative 1 end subscript superscript 1 3 over 2 left parenthesis straight x plus 1 right parenthesis space dx plus integral subscript 1 superscript 3 open parentheses negative 1 half straight x plus 7 over 2 close parentheses dx space minus space integral subscript negative 1 end subscript superscript 3 open parentheses 1 half straight x plus 1 half close parentheses dx
equals space 3 over 2 open square brackets straight x squared over 2 plus straight x close square brackets subscript negative 1 end subscript superscript 1 plus open square brackets negative straight x squared over 4 plus 7 over 2 straight x close square brackets subscript 1 superscript 3 space minus space open square brackets straight x squared over 4 plus straight x over 2 close square brackets subscript negative 1 end subscript superscript 3
equals space 3 over 2 open square brackets open parentheses 1 half plus 1 close parentheses minus open parentheses 1 half minus 1 close parentheses close square brackets plus open square brackets open parentheses negative 9 over 4 plus 21 over 2 close parentheses minus open parentheses negative 1 fourth plus 7 over 2 close parentheses close square brackets minus open square brackets open parentheses 9 over 4 plus 3 over 2 close parentheses minus open parentheses 1 fourth minus 1 half close parentheses close square brackets
equals space 3 over 2 open parentheses 3 over 2 plus 1 half close parentheses plus open parentheses 33 over 4 minus 13 over 4 close parentheses minus open parentheses 15 over 4 plus 1 fourth close parentheses
equals space 9 over 4 plus 3 over 4 plus 33 over 4 minus 13 over 4 minus 15 over 4 minus 1 fourth equals 16 over 4 space equals space 4 space sq. space units.

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