If the area of the region (x,y):|x-5| y 4x is A, then 3A is equal to _________.

Question

If the area of the region ${(x,y):|x-5|\le y\le 4\sqrt{x}}$ is $A$, then $3A$ is equal to \_\_\_\_\_\_\_\_\_.

Answer: 368

Accepted Answer

The lines $y=5-x$ (for $x\le5$) and $y=x-5$ (for $x\ge5$) meet $y=4\sqrt x$ at $x=1$ and $x=25$ respectively.

$$A = \int_1^{25}4\sqrt{x}\,dx - \frac12(5-1)(4) - \frac12(25-5)(20)$$

$$= 4\left[\frac23 x^{3/2}\right]_1^{25} - 8 - 200 = 4\cdot\frac23(125-1) - 208 = \frac{992}{3}-208 = \frac{368}{3}$$

$3A = 368$

Solutions · 1