Question

Let the area enclosed by the curve $y=1-x^2$ and the line $y=a$, where $0 \leq a<1$, be represented byA(a). If $\frac{ A (0)}{ A \left(\frac{1}{2}\right)}= k$, then

• A. $1< k <\frac{3}{2}$
• B. $\frac{3}{2}< k <2$
• C. $2< k <\frac{5}{2}$
• D. $ \frac{5}{2}< k <3$

Answer 1

Answer: (D)

$A(a)=2 \int\limits_0^{\sqrt{1-4}}\left(\left(1-x^2\right)-a\right) d x=\frac{4}{3}(1-a)^{3 / 2}$
$\therefore A (0)=\frac{4}{3}$
and $A \left(\frac{1}{2}\right)=\frac{4}{3}\left(\frac{1}{2}\right)^{\frac{3}{2}} \Rightarrow \frac{ A (0)}{ A \left(\frac{1}{2}\right)}=2 \sqrt{2}$