If the area of the region bounded by the curves y=4-(x^2)/(4) and y=(x-4)/(2) is equal to …

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 $\alpha$, then $6 \alpha$. equals

Accepted Answer

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

$$\begin{aligned}
& \text { Area }=\int_\limits{-6}^4\left(4-\frac{x^2}{4}-\frac{x}{2}+2\right) d x \
& =\left[6 x-\frac{x^3}{12}-\frac{x^2}{4}\right]_{-6}^4 \
& =6(4+6)-\left(\frac{64}{12}+\frac{216}{12}\right)-\left(\frac{16}{4}-\frac{36}{4}\right) \
& =60-\frac{70}{3}+5 \
& \alpha=\frac{125}{3} \
& 6 \alpha=6 \times \frac{125}{3}=250
\end{aligned}$$

Solutions · 1