JEE MAIN - Mathematics (2020 - 3rd September Morning Slot - No. 14)
The area (in sq. units) of the region
{ (x, y) : 0 $$ \le $$ y $$ \le $$ x2 + 1, 0 $$ \le $$ y $$ \le $$ x + 1,
$${1 \over 2}$$ $$ \le $$ x $$ \le $$ 2 } is :
{ (x, y) : 0 $$ \le $$ y $$ \le $$ x2 + 1, 0 $$ \le $$ y $$ \le $$ x + 1,
$${1 \over 2}$$ $$ \le $$ x $$ \le $$ 2 } is :
$${{79} \over {16}}$$
$${{79} \over {24}}$$
$${{23} \over {6}}$$
$${{23} \over {16}}$$
Explanation
_3rd_September_Morning_Slot_en_14_1.png)
$$A = \int\limits_{{1 \over 2}}^1 {({x^2} + 1)dx + } $$$$\int\limits_1^2 {} $$$${(x + 1)dx}$$= $${\left[ {{{{x^3}} \over 3} + x} \right]_{{1 \over 2}}^1 + \left[ {{{{x^2}} \over 2} + x} \right]_1^2}$$
$${ = \left( {{4 \over 3} - {{13} \over {24}}} \right) + \left( {4 - {3 \over 2}} \right)}$$
$${ = {{19} \over {24}} + {5 \over 2}}$$
$${ = {{79} \over {24}}}$$
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