Question

Let the area enclosed by the curve $y=1-x^2$ and the line $y=a$, where $0\le a<1$, 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)

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