3. the graph of a function f(x) is depicted below. if the area of the shaded region labeled a is 0.71 and…

3. the graph of a function f(x) is depicted below. if the area of the shaded region labeled a is 0.71 and the area of the shaded region labeled b is 0.34 then ∫₀²f(x)dx must be... a ) 1.05 b ) -0.31 c ) 0.95 d ) -0.37

3. the graph of a function f(x) is depicted below. if the area of the shaded region labeled a is 0.71 and the area of the shaded region labeled b is 0.34 then ∫₀²f(x)dx must be... a ) 1.05 b ) -0.31 c ) 0.95 d ) -0.37

Answer

Explanation:

Step1: Recall definite - integral property

The definite integral $\int_{0}^{2}f(x)dx$ is the net - signed area between the curve $y = f(x)$ and the $x$ - axis from $x = 0$ to $x = 2$. Areas below the $x$ - axis contribute negatively and areas above the $x$ - axis contribute positively.

Step2: Analyze the areas of regions A and B

Region A is below the $x$ - axis, so its contribution to the integral is negative, and region B is above the $x$ - axis, so its contribution is positive. Let $A_{A}=0.71$ (area of region A) and $A_{B}=0.34$ (area of region B). Then $\int_{0}^{2}f(x)dx=A_{B}-A_{A}$.

Step3: Calculate the value of the integral

Substitute the values of $A_{A}$ and $A_{B}$ into the formula: $\int_{0}^{2}f(x)dx = 0.34−0.71=- 0.37$.

Answer:

D. -0.37