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

Explanation:

Step1: Divide the region

Divide the area under the curve from (x = 0) to (x=10) into geometric - shapes. We have a triangle from (x = 0) to (x = 3), a triangle from (x = 3) to (x = 4), and a rectangle from (x = 4) to (x = 8), and a triangle from (x = 8) to (x = 10).

Step2: Calculate area of first triangle

The first triangle from (x = 0) to (x = 3) has base (b_1=3) and height (h_1 = 6). The area of a triangle is (A=\frac{1}{2}bh). So (A_1=\frac{1}{2}\times3\times6 = 9). But since it is below the (x) - axis, (A_1=- 9).

Step3: Calculate area of second triangle

The second triangle from (x = 3) to (x = 4) has base (b_2 = 1) and height (h_2=6). So (A_2=\frac{1}{2}\times1\times6 = 3).

Step4: Calculate area of rectangle

The rectangle from (x = 4) to (x = 8) has base (b_3=4) and height (h_3 = 1). The area of a rectangle is (A = bh), so (A_3=4\times1=4).

Step5: Calculate area of third triangle

The third triangle from (x = 8) to (x = 10) has base (b_4 = 2) and height (h_4=1). So (A_4=\frac{1}{2}\times2\times1 = 1).

Step6: Sum up the areas

(\int_{0}^{10}f(x)dx=A_1 + A_2+A_3+A_4=-9 + 3+4 + 1=-1).

Answer:

(-1)