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. $e^{a^2}+1sq.unit$
• D. $e^{a^2}-1sq.unit$

Answer 1

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

Required area is ${\int }_0^{a}ydx={\int }_0^{a}xe{x}^{2}dx$
$Weput{x}^2=t\Rightarrow dx=\frac{dt}{2x}asx=0\Rightarrow t=0and$
$x=a\Rightarrow t=a^2,$ then it reduces to
$\frac{1}{2}{\int }_0^{a^2}{e}^{t}dt=\frac{1}{2}[e^t{]}_0^{a^2}=\frac{e^{a^2}-1}{2}sq.unit$