JEE MAIN - Mathematics (2024 - 4th April Evening Shift - No. 8)
The area (in sq. units) of the region described by $$
\left\{(x, y): y^2 \leq 2 x \text {, and } y \geq 4 x-1\right\}
$$ is
$$\frac{9}{32}$$
$$\frac{11}{12}$$
$$\frac{8}{9}$$
$$\frac{11}{32}$$
Explanation
$$\text { Area }=\int_\limits{-\frac{1}{2}}^1\left(\frac{y+1}{4}-\frac{y^2}{2}\right) d y$$
_4th_April_Evening_Shift_en_8_1.png)
$$\begin{aligned} & =\left[\frac{y^2}{8}+\frac{y}{4}-\frac{y^3}{6}\right]_{-\frac{1}{2}}^1 \\ & =\left(\frac{1}{8}+\frac{1}{4}-\frac{1}{6}\right)-\left(\frac{1}{32}-\frac{1}{8}+\frac{1}{48}\right) \\ & =\frac{5}{24}+\frac{7}{96} \\ & =\frac{27}{96} \\ & =\frac{9}{32} \end{aligned}$$
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