question 2 of 13 using the figure below, decide whether each of the following definite integrals is positive…

Explanation:

Step1: Recall integral - area relationship

The definite integral $\int_{a}^{b}f(x)dx$ is 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 contribute positively and areas below the $x$ - axis contribute negatively.

Step2: Analyze $\int_{-5}^{-4}f(x)dx$

The function $y = f(x)$ is below the $x$ - axis for $x\in[-5,-4]$. So, $\int_{-5}^{-4}f(x)dx<0$.

Step3: Analyze $\int_{-4}^{1}f(x)dx$

The area above the $x$ - axis from $x=-4$ to some point and the area below the $x$ - axis from that point to $x = 1$ need to be considered. The positive area (above $x$ - axis) is larger than the negative area (below $x$ - axis) in the interval $[-4,1]$. So, $\int_{-4}^{1}f(x)dx>0$.

Step4: Analyze $\int_{1}^{3}f(x)dx$

The function $y = f(x)$ is below the $x$ - axis for $x\in[1,3]$. So, $\int_{1}^{3}f(x)dx<0$.

Step5: Analyze $\int_{-5}^{3}f(x)dx$

The positive areas above the $x$ - axis and negative areas below the $x$ - axis in the interval $[-5,3]$ need to be considered. The positive areas are larger than the negative areas. So, $\int_{-5}^{3}f(x)dx>0$.

Answer:

(a) Negative (b) Positive (c) Negative (d) Positive