JEE 2024 — Mathematics PYQ

\begin{aligned}
 & \text{Required area} \\
 & = \int_{-2}^{-1}\frac{x+4}{2}dx+\int_{-1}^{8}\frac{\sqrt{8-x}}{2}dx-\frac{1}{\sqrt{2}}\int_{-2}^{0}\sqrt{x}dx \\
 & =\frac{1}{2}\biggl(\frac{x^{2}}{2}+4x\biggr)\biggr|_{2}^{-1}+\frac{1}{2}\biggl[\frac{-2}{3}(8-x)^{\frac{3}{2}}\biggr]_{-1}^{8}+\frac{1}{\sqrt{2}}\times\frac{2}{3}(-x)^{\frac{3}{2}}\biggr]_{-2}^{0} \\
 & =\frac{1}{2}\biggl[\frac{1}{2}-4-2+8\biggr]-\frac{1}{3}\biggl[(0)-(9)^{\frac{3}{2}}\biggr]+\frac{\sqrt{2}}{3}\biggl[0-2\sqrt{2}\biggr] \\
 & =\frac{107}{12}=\frac{m}{n} \\
 & \Rightarrow m+n=107+12=119
\end{aligned}