Question

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

• 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

Answer 1

Answer: (B)

$\frac{e^{a^2}-1}{2}$ sq. unit


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