CBSE 2022 · Region 4 · Set 3 · Q7 · 3 marks
If the area of the region bounded by the line $\displaystyle \mathrm{y}=\mathrm{mx}$ and the curve $\displaystyle \mathrm{x}^{2}=\mathrm{y}$ is $\displaystyle \frac{32}{3}$ sq. units, then find the positive value of m , using integration.
Marking-scheme solution
\(\displaystyle x\)-coordinates of points of intersection are \(\displaystyle 0, m\).
\[\begin{aligned}
& \text { According to question }=\int_{0}^{m}\left(m x-x^{2}\right) d x=\frac{3}{3} \\
& \Rightarrow m \frac{x^{2}}{2}-\left.\frac{x^{3}}{3}\right|_{0} ^{m}=\frac{32}{3} \\
& \frac{m^{3}}{6}=\frac{32}{3} \Rightarrow m^{3}=64 \\
& \Rightarrow m=4
\end{aligned}
\]
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