If the area of the region bounded by the curves and is equal to , then equals [2025]

Question

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 $α$ , then $6 α$ equals [2025]

Smart Achievers · Accepted Answer

(4)

Given, $y = 4 - \frac{x^{2}}{4}$ ... (i)

and $y = \frac{x - 4}{2}$ ... (ii)

From (i) & (ii), we get

x = –6, 4 and y = –5, 0

Thus, point of intersection of (i) & (ii) are (–6, 5) and (4, 0).

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

$\Rightarrow α = \int_{- 6}^{4} {- \frac{x^{2}}{4} - \frac{x}{2} + 6} d x$

$α =$ $- \frac{x^{3}}{12} - \frac{x^{2}}{4} + 6 x |_{- 6}^{4}$

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

$= \frac{125}{3}$

$∴ 6 α = 250$ .

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