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

Explanation:

Step1: Divide the region under the curve

The region between (x = 2) and (x=10) under (y = f(x)) can be divided into two triangles and a 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), (b_2=1) and height (h = 2) (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)\times2}{2}=3).

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 + 3+2=8).

Answer:

8