Mathematics - Area Under Curves Question with Solution | TestHub

Let, the parabola be $y=a{x}^2+bx+c$,

Now given parabola is passing through $\left(-1,0\right), \left(0,1\right) \& \left(1,0\right)$

Now putting the value in the above equation of parabola we get,

$a-b+c=0 ......\left(1\right)$, $c=1 ....\left(2\right)$ and $a+b+c=0 ....\left(3\right)$

Now solving above three equations we get,

Parabola as $y=1-x^2$

Now area of region between $y=1-x^2$ and ${\left(x+1\right)}^2+{\left(y-1\right)}^2\le 1$ is given by,

Now from above diagram, required area will be,

$A={\int }_{-1}^0\left(-x^2+\sqrt{1-{\left(x+1\right)}^2}\right)dx$

$\Rightarrow A=\frac{\pi }{4}-\frac{1}{3}$

Hence, $12(\pi -4A)=16$