12 the graph of y = f(x) consists of a semi - circle with endpoints at (2, - 6) and (12, - 6), as shown in…

Explanation:

Step1: Determine the radius of the semi - circle

The diameter of the semi - circle is the distance between (x = 2) and (x=12). So (d=12 - 2=10), and the radius (r = 5).

Step2: Calculate the area of the semi - circle

The area of a full - circle is (A=\pi r^{2}), so the area of a semi - circle is (A_{semicircle}=\frac{1}{2}\pi r^{2}=\frac{1}{2}\pi(5)^{2}=\frac{25\pi}{2}).

Step3: Consider the region below the x - axis

The semi - circle is below the (x) - axis. The definite integral (\int_{2}^{12}f(x)dx) gives the net signed area between the curve (y = f(x)) and the (x) - axis. Since the region is below the (x) - axis, the value of the integral is the negative of the area of the semi - circle. So (\int_{2}^{12}f(x)dx=-\frac{25\pi}{2}).

Answer:

C. (\frac{-25\pi}{2})