for the function f whose graph is given below, list the following quantities in increasing order, from least…

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 $\int_{0}^{5}f(x)dx$ (D)

From $x = 0$ to $x = 5$, the function $y=f(x)$ is below the $x$ - axis. So, $\int_{0}^{5}f(x)dx<0$.

Step3: Analyze $\int_{0}^{8}f(x)dx$ (B)

$\int_{0}^{8}f(x)dx=\int_{0}^{5}f(x)dx+\int_{5}^{8}f(x)dx$. Since $\int_{0}^{5}f(x)dx<0$ and $\int_{5}^{8}f(x)dx>0$ (the part from $x = 5$ to $x = 8$ is above the $x$ - axis), but the negative part from $0$ to $5$ may still make $\int_{0}^{8}f(x)dx<0$ or positive depending on the magnitudes of the two areas.

Step4: Analyze $\int_{3}^{8}f(x)dx$ (A)

$\int_{3}^{8}f(x)dx=\int_{3}^{5}f(x)dx+\int_{5}^{8}f(x)dx$. The part $\int_{3}^{5}f(x)dx<0$ and $\int_{5}^{8}f(x)dx>0$. Since the negative part from $3$ to $5$ is smaller in magnitude than the positive part from $5$ to $8$ (by observing the graph), $\int_{3}^{8}f(x)dx>0$.

Step5: Analyze $\int_{5}^{8}f(x)dx$ (C)

The function $y = f(x)$ is above the $x$ - axis from $x = 5$ to $x = 8$, so $\int_{5}^{8}f(x)dx>0$. Also, $\int_{5}^{8}f(x)dx$ is a part of $\int_{3}^{8}f(x)dx$, so $\int_{5}^{8}f(x)dx<\int_{3}^{8}f(x)dx$.

Step6: Compare the values

We know that $\int_{0}^{5}f(x)dx$ is negative. $\int_{0}^{8}f(x)dx$ is the sum of a negative ($\int_{0}^{5}f(x)dx$) and a positive ($\int_{5}^{8}f(x)dx$) value. $\int_{3}^{8}f(x)dx$ is positive and $\int_{5}^{8}f(x)dx$ is positive and smaller than $\int_{3}^{8}f(x)dx$. By observing the graph, we can see that $\int_{0}^{5}f(x)dx$ is the most negative, then $\int_{0}^{8}f(x)dx$ (because it includes the negative part from $0$ to $5$), then $\int_{5}^{8}f(x)dx$, and finally $\int_{3}^{8}f(x)dx$ is the largest.

Answer:

D,B,C,A