QJEEM23Ma6a1d: JEE Main 2023 Mathematics - Class 12 JEEM23Ma6a1d Solution: The area of the region is :

Question

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The area of the region $\left\{(x, y): x^{2} \leq y \leq 8-x^{2}, y \leq 7\right\}$ is :

Accepted Answer

The given curves are

x^2 \leq y, y \leq 8-x^2 ; y \leq 7

On solving, we get

x^2=8-x^2

\begin{aligned}&amp; \Rightarrow x^2=4 \\\\&amp; \Rightarrow x= \pm 2\end{aligned}

\text { So, area }=2\left[\int_0^4 \sqrt{y} d y+\int_4^7 \sqrt{8-y} d y\right]

=2\left\{\left[\frac{y^{\frac{3}{2}}}{\frac{3}{2}}\right]_0^4+\left[\frac{-(8-y)^{\frac{3}{2}}}{\frac{3}{2}}\right]_4^7\right\}

\begin{aligned}&amp; =2 \times \frac{2}{3}\left\{\left[4^{3 / 2}-0\right]+\left(-(1)^{3 / 2}+(4)^{3 / 2}\right)\right\} \\\\&amp; =\frac{4}{3}\{8-1+8\}=\frac{4}{3} \times 15=20 \text { sq. units }\end{aligned}

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