Question

score on last try: 2 of 4 pts. see details for more. at least one scored part is incorrect. jump to first changable incorrect part. next question get a similar question you can retry this question below to find the blue shaded area above, we would calculate: \\(\int_{a}^{b} f(x) dx = \text{area}\\) where: \\(a = 1\\), \\(b = 2\\) \\(f(x) = \\) \\(\text{area} = \\) (hints: you learned to find the equation of a line in algebra and the area of a trapezoid in high school!) question help: video

Explanation:

Step1: Find line equation slope

The line passes through $(0,4)$ and $(4,2)$.
Slope $m = \frac{2-4}{4-0} = \frac{-2}{4} = -\frac{1}{2}$

Step2: Define $f(x)$ using intercept

The y-intercept is 4, so using $y=mx+b$:
$f(x) = -\frac{1}{2}x + 4$

Step3: Calculate area via trapezoid formula

The shaded region is a trapezoid with bases $f(1)=3.5$ and $f(2)=3$, height $2-1=1$.
Area $= \frac{1}{2} \times (f(1)+f(2)) \times (b-a)$
$= \frac{1}{2} \times (3.5 + 3) \times 1$

Answer:

$f(x) = -\frac{1}{2}x + 4$
$\text{area} = \frac{13}{4}$ (or 3.25)