the graph of f is shown. evaluate each integral by interpreting it in terms of areas. (a) ∫₀² f(x) dx (b)…

Explanation:

Step1: Analyze integral (a)

The region from (x = 0) to (x=2) under (y = f(x)) is a trapezoid. The formula for the area of a trapezoid is (A=\frac{(b_1 + b_2)h}{2}), where (b_1) and (b_2) are the lengths of the parallel - sides and (h) is the height. Here, (b_1 = 1), (b_2 = 3), and (h = 2). So (A=\frac{(1 + 3)\times2}{2}=4).

Step2: Analyze integral (b)

The region from (x = 0) to (x = 5) is composed of the trapezoid from (x = 0) to (x = 2) (area (A_1=4)) and a triangle from (x = 2) to (x = 5). The triangle has base (b = 3) and height (h = 4). The area of the triangle is (A_2=\frac{1}{2}\times3\times4 = 6). So (\int_{0}^{5}f(x)dx=4 + 6=10).

Step3: Analyze integral (c)

The region from (x = 5) to (x = 7) is a triangle below the (x) - axis. The base (b = 2) and height (h = 3). The area of the triangle is (A=\frac{1}{2}\times2\times3=3), but since it is below the (x) - axis, (\int_{5}^{7}f(x)dx=- 3).

Step4: Analyze integral (d)

(\int_{0}^{9}f(x)dx=\int_{0}^{5}f(x)dx+\int_{5}^{7}f(x)dx+\int_{7}^{9}f(x)dx). We know (\int_{0}^{5}f(x)dx = 10), (\int_{5}^{7}f(x)dx=-3). The region from (x = 7) to (x = 9) is a trapezoid with (b_1 = 1), (b_2 = 3), and (h = 2), so its area (A_3=\frac{(1 + 3)\times2}{2}=4). Then (\int_{0}^{9}f(x)dx=10-3 + 4=11).

Answer:

11