calculate the definite integral ∫d to b f(x) dx by referring to the figure on the right with the indicated…

Explanation:

Step1: Recall definite - integral property

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

Step2: Analyze the areas in the figure

From $x = b$ to $x = d$, area $B$ is below the $x$ - axis (negative) and area $C$ is above the $x$ - axis (positive) and area $D$ is below the $x$ - axis (negative). So, $\int_{b}^{d}f(x)dx=- \text{Area}(B)+\text{Area}(C)-\text{Area}(D)$.

Step3: Substitute the given values

We know that $\text{Area}(B) = 2450$, $\text{Area}(C)=3173$, and $\text{Area}(D)=1615$. Then $\int_{b}^{d}f(x)dx=-2450 + 3173-1615$.

Step4: Calculate the result

$-2450+3173 - 1615=3173-(2450 + 1615)=3173 - 4065=-892$. Since $\int_{d}^{b}f(x)dx=-\int_{b}^{d}f(x)dx$, then $\int_{d}^{b}f(x)dx = 892$.

Answer:

$892$