Question

The area enclosed by the polynomial function of least degree satisfying ${lim}_{x\to 0}{[1+\frac{f\left(x\right)}{x^3}]}^{\frac{1}{x}}=e$ and the circle $x^2+y^2=2$ above the X-axis is ______ in square units.

• A. $\frac{3}{5}+\frac{\pi }{2}$
• B. $\frac{2}{5}-\frac{\pi }{2}$
• C. $\frac{2}{5}+\frac{\pi }{2}$
• D. $\frac{1}{5}+\frac{\pi }{2}$

Answer 1

The correct option is A $\frac{3}{5}+\frac{\pi }{2}$

$e^{\underset{x\to \infty }{lim}\frac{1}{x}[\frac{f\left(x\right)}{x^3}]}=e⟹f\left(x\right)=x^4$

$a={\int }_0^1(x-x^4)dx=\frac{3}{10};b=\frac{1}{8}\pi \left(2\right)=\frac{\pi }{4}⟹area=2a+2b=\frac{3}{5}+\frac{\pi }{2}$