Question

evaluate the definite integral by interpreting it in terms of areas.\\(\int_{2}^{9} (2x - 16)dx\\)

Explanation:

Step1: Find x-intercept of $2x-16$

Set $2x-16=0$, solve for $x$:
$$2x=16 \implies x=8$$

Step2: Split integral at $x=8$

The integral splits into two parts:
$$\int_{2}^{9}(2x-16)dx = \int_{2}^{8}(2x-16)dx + \int_{8}^{9}(2x-16)dx$$

Step3: Calculate area for $\int_{2}^{8}(2x-16)dx$

This is a triangle below x-axis. At $x=2$, $2(2)-16=-12$. Base length: $8-2=6$, height: $12$. Area (negative for below axis):
$$-\frac{1}{2} \times 6 \times 12 = -36$$

Step4: Calculate area for $\int_{8}^{9}(2x-16)dx$

This is a triangle above x-axis. At $x=9$, $2(9)-16=2$. Base length: $9-8=1$, height: $2$. Area:
$$\frac{1}{2} \times 1 \times 2 = 1$$

Step5: Sum the two results

Add the two computed areas:
$$-36 + 1 = -35$$

Answer:

$-35$