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Application of Integrals

Name: Zigya - For The Curious Learner
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Short 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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Long 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).

The equation of AB is
   $straight y minus 0 space space equals space fraction numerator 5 minus 0 over denominator 4 minus 2 end fraction left parenthesis straight x minus 2 right parenthesis$
or    $straight y space equals space 5 over 2 left parenthesis straight x minus 2 right parenthesis$
The equation of BC is
$straight y minus 5 space equals fraction numerator 3 minus 5 over denominator 6 minus 4 end fraction left parenthesis straight x minus 4 right parenthesis$
or    $straight y minus 5 space equals space minus left parenthesis straight x minus 4 right parenthesis space space or space space space straight y space equals space minus straight x plus 9$
The equation of CA is
$straight y minus 3 space equals space fraction numerator 0 minus 3 over denominator 2 minus 6 end fraction left parenthesis straight x minus 6 right parenthesis space space or space straight y minus 3 space equals space 3 over 4 left parenthesis straight x minus 6 right parenthesis space or space space space straight y space equals space fraction numerator 3 straight x over denominator 4 end fraction minus 3 over 2$

From C, draw CD $perpendicular$ x-axis.
Required area = area of quad. ABCD - area of $increment space ADC$
   $equals space open square brackets integral subscript 2 superscript 4 5 over 2 left parenthesis straight x minus 2 right parenthesis space dx space plus space integral subscript 4 superscript 6 left parenthesis negative straight x plus 9 right parenthesis space dx close square brackets space minus space integral subscript 2 superscript 6 open parentheses fraction numerator 3 straight x over denominator 4 end fraction minus 3 over 2 close parentheses dx equals space 5 over 2 open square brackets straight x squared over 2 minus 2 straight x close square brackets subscript 2 superscript 4 space plus open square brackets negative straight x squared over 2 plus 9 straight x close square brackets subscript 4 superscript 6 space minus space open square brackets fraction numerator 3 straight x squared over denominator 8 end fraction minus 3 over 2 straight x close square brackets subscript 2 superscript 6 equals space 5 over 2 open square brackets left parenthesis 8 minus 8 right parenthesis minus left parenthesis 2 minus 4 right parenthesis close square brackets plus open square brackets left parenthesis negative 18 plus 54 right parenthesis minus left parenthesis negative 8 plus 36 right parenthesis close square brackets minus open square brackets open parentheses 27 over 2 minus 9 close parentheses minus open parentheses 3 over 2 minus 3 close parentheses close square brackets equals space 5 over 2 left parenthesis 0 plus 2 right parenthesis space plus space left parenthesis 36 minus 28 right parenthesis space minus open parentheses 9 over 2 plus 3 over 2 close parentheses equals space 5 plus 8 minus 6 space equals space 7 space sq. space units.$