Question

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

Solution

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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Vector Algebra

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

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