If the area of the region bounded by the curves y=4-frac x 2 4 and y=x-4/2 is equal to α , then 6 α equals

Mathematics · Area under the Curves

JEE Main 2025 — 7 April, Morning Shift — Question 24

If the area of the region bounded by the curves $y=4-\frac{x^{2}}{4}$ and $y=\frac{x-4}{2}$ is equal to $\alpha$ , then $6 \alpha$ equals

$y=4-\frac{x^{2}}{4}$ and $y=\frac{x-4}{2}$

Area $=\int_{-6}^{4}\left(4-\frac{x^{2}}{4}-\frac{x}{2}+2\right) d x$

$=\left[6 x-\frac{x^{3}}{12}-\frac{x^{2}}{4}\right]_{-6}^{4}$

$=6(4+6)-\left(\frac{64}{12}+\frac{216}{12}\right)-\left(\frac{16}{4}-\frac{36}{4}\right)$

$=60-\frac{70}{3}+5$

$\alpha=\frac{125}{3}$

$6 \alpha=6 \times \frac{125}{3}=250$

Answer key and solution verified before publishing.

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