question the piecewise function f(x) is graphed below. use geometric formulas to evaluate the following…

Explanation:

Step1: Divide the region

The region under the curve from $x = 2$ to $x=10$ can be divided into two triangles and one trapezoid.

Step2: Calculate area of first - triangle

The first triangle has base $b_1=3$ (from $x = 2$ to $x = 5$) and height $h_1 = 2$. Using the formula for the area of a triangle $A=\frac{1}{2}bh$, we have $A_1=\frac{1}{2}\times3\times2 = 3$.

Step3: Calculate area of trapezoid

The trapezoid has bases $b_1 = 2$ and $b_2=1$ and height $h = 3$ (from $x = 5$ to $x = 6$). Using the formula for the area of a trapezoid $A=\frac{(b_1 + b_2)h}{2}$, we get $A_2=\frac{(2 + 1)\times3}{2}=\frac{9}{2}$.

Step4: Calculate area of second - triangle

The second triangle has base $b_3=4$ (from $x = 6$ to $x = 10$) and height $h_3=1$. Using the formula for the area of a triangle $A=\frac{1}{2}bh$, we have $A_3=\frac{1}{2}\times4\times1 = 2$.

Step5: Calculate the definite - integral

The definite integral $\int_{2}^{10}f(x)dx$ is the sum of the areas of these geometric shapes. So $\int_{2}^{10}f(x)dx=A_1+A_2+A_3=3+\frac{9}{2}+2=\frac{6 + 9+4}{2}=\frac{19}{2}$.

Answer:

$\frac{19}{2}$