Question

each of the regions a, b, and c enclosed by the graph of f and the x - axis has area 6. find the value of each of the following definite integrals. (a) $int_{-4}^{0}f(x)dx$ (b) $int_{-4}^{2}f(x)dx$

Explanation:

Step1: Interpret integral as area

The definite - integral $\int_{a}^{b}f(x)dx$ represents the net - signed area between the curve $y = f(x)$ and the $x$ - axis from $x=a$ to $x = b$. Areas above the $x$ - axis are positive and areas below the $x$ - axis are negative.

Step2: Analyze $\int_{-4}^{0}f(x)dx$

The interval $[-4,0]$ covers regions A and B. Region A is above the $x$ - axis and region B is below the $x$ - axis. Since each region has an area of 6, $\int_{-4}^{0}f(x)dx=6+( - 6)=0$.

Step3: Analyze $\int_{-4}^{2}f(x)dx$

The interval $[-4,2]$ covers regions A, B, and C. Region A is above the $x$ - axis ($+6$), region B is below the $x$ - axis ($-6$), and region C is below the $x$ - axis ($-6$). So, $\int_{-4}^{2}f(x)dx=6+( - 6)+( - 6)=-6$.

Answer:

(a) 0
(b) - 6