A) \[\frac{{{e}^{{{a}^{2}}}}+1}{2}\]sq. unit
B) \[\frac{{{e}^{{{a}^{2}}}}-1}{2}\]sq. unit
C) \[{{e}^{{{a}^{2}}}}+1\]sq. unit
D) \[{{e}^{{{a}^{2}}}}-1\]sq. unit
Correct Answer: B
Solution :
Required area is \[\int_{0}^{a}{y\,\,dx=\int_{0}^{a}{x{{e}^{{{x}^{2}}}}dx}}\] We put \[{{x}^{2}}=t\Rightarrow dx=\frac{dt}{2x}\] as \[x=0\Rightarrow t=0\] and \[x=a\Rightarrow t={{a}^{2}}\], then it reduces to \[\frac{1}{2}\int_{0}^{{{a}^{2}}}{{{e}^{t}}dt=\frac{1}{2}[{{e}^{t}}]_{0}^{{{a}^{2}}}=\frac{{{e}^{{{a}^{2}}}}-1}{2}}\] sq. unit.
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