The area of the region described by A = (x,y) : x^2+y^21 and y^21-x is:

Question

The area of the region described by $A = \{(x,y) : x^2+y^2\le1$ and $y^2\le1-x\}$ is:

Accepted Answer

Given curves are ${x^2} + {y^2} = 1$ and ${y^2} = 1 - x.$

Intersection points are $x = 0,1$

Area of shaded portion is the required area.

So,  Required Area $=$ Area of semi-circle $+$ Area bounded by parabola

$$ = {{\pi {r^2}} \over 2} + 2\int\limits_0^1 {\sqrt {1 - x} dx} $$

$$ = {\pi  \over 2} + 2\int\limits_0^1 {\sqrt {1 - x} \,dx} $$

( As radius of circle $=1$ )

$$ = {\pi  \over 2} + 2\left[ {{{{{\left( {1 - x} \right)}^{{\raise0.5ex\hbox{$\scriptstyle 3$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 2$}}}}} \over { - {\raise0.5ex\hbox{$\scriptstyle 3$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 2$}}}}} \right]_0^1$$

$$ = {\pi  \over 2} - {4 \over 3}\left( { - 1} \right) = {\pi  \over 2} + {4 \over 3}$$ Sq. unit

Solutions · 1