The area (in sq.

Question

The area (in sq. units) of the region described by $A=\{(x,y)\mid y\ge x^2-5x+4,\ x+y\ge1,\ y\le0\}$ is:

Accepted Answer

The parabola $y=x^2-5x+4$ meets $y=0$ at $x=1,4$; combined with $x+y\ge1$, the relevant intersection with the line ($y=0$ boundary) occurs at $x=1$ and the parabola crosses again at $x=4$ giving the strip $[3,4]$ where the constraints jointly bind (per source figure).

$$\text{Area} = \frac12\times2\times2 + \int_3^4(x^2-5x+4)\,dx = 2+\frac76 = \frac{19}{6}$$

Solutions · 1