QJEEM16Macaed: JEE Main 2016 Mathematics - Class 12 JEEM16Macaed Solution: The area (in sq. units) of the region described by A= {(x, y) y …

Question

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The area (in sq. units) of the region described by

A= {(x, y)

\left| {} \right.

y

\ge

x²

-

5x + 4, x + y

\ge

1, y

\le

0} is :

Accepted Answer

Required Area

= A₁ + A₂

=

\left| {\int\limits_1^3 {\left( {1 - x} \right)} dx} \right| + \left| {\int\limits_3^4 {\left( {{x^2} - 5x + 4} \right)dx} } \right|

=

\left| {\left[ {x - {{{x^2}} \over 2}} \right]_1^3} \right| + \left| {\left[ {{{{x^3}} \over 3} - {5 \over 2}{x^2} + 4x} \right]_3^4} \right|

=

\left| {\left[ {\left( {3 - {9 \over 2}} \right) - \left( {1 - {1 \over 2}} \right)} \right]} \right| + \left| {\left[ {\left( {{{64} \over 3} - 40 + 16} \right) - \left( {9 - {{45} \over 2} + 12} \right)} \right]} \right|

=

\left| {\left( {2 - 4} \right)} \right| + \left| {\left( {{{ - 8} \over 3} + {3 \over 2}} \right)} \right|

= 2 +

{7 \over 6}

=

{{19} \over 6}

sq. unit.

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