Question

Area bounded by the curve $y=x{e}^{x^2}$ with the $x$-axis between the ordinates $x=0$ and $x=a$

• A. $\frac{e^{a^2}+1}{2}$ sq. units
• B. $\frac{e^{a^2}-1}{2}$sq. units
• C. $e^{a^2}-1$ sq. units
• D. $e^{a^2}+1$ sq. units

Answer 1

The correct option is B $\frac{e^{a^2}-1}{2}$sq. units

Required area is given by
${\int }_0^{a}ydx={\int }_0^{a}x{e}^{x^2}dx$

Put $x^2=t\Rightarrow dx=\frac{dt}{2x}$
$x=0\to t=0$ and $x=a\to t=a^2$

Then the given integral reduces to
$\frac{1}{2}{\int }_0^{a^2}{e}^{t}dt=\frac{1}{2}|e^t{|}_0^{a^2}=\frac{e^{a^2}-1}{2}$