The area of the region [(x, y): y^2 4 x, x0, x 3] is

Question

The area of the region $$\left\{(x, y): y^2 \leq 4 x, x0, x \neq 3\right\}$$ is

Accepted Answer

$$\begin{aligned} & y^2 \leq 4 x, x<4 \ & \frac{x y(x-1)(x-2)}{(x-3)(x-4)}>0 \ & \text { Case - I}: y>0 \ & \frac{x(x-1)(x-2)}{(x-3)(x-4)}>0 \ & x \in(0,1) \cup(2,3) \ & \text { Case }- \text { II : y<0 } \ & \frac{x(x-1)(x-2)}{(x-3)(x-4)}<0, x \in(1,2) \cup(3,4) \end{aligned}$$ $$\begin{aligned} & \text { Area }=2 \int_\limits0^4 \sqrt{x} d x \ & =2 \cdot \frac{2}{3}\left[x^{3 / 2}\right]_0^4=\frac{32}{3} \end{aligned}$$

Solutions · 1