$g(0)$ requires reversing the limits of integration:$$g(0) = \int_{6}^{0} f(t) dt = - \int_{0}^{6} f(t) dt$$The integral $\int_{0}^{6} f(t) dt$ is the area of the semicircle above the $x$-axis with radius $r=3$.$$\int_{0}^{6} f(t) dt = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (3)^2 = \frac{9\pi}{2}$$$$\mathbf{g(0) = - \frac{9\pi}{2}}$$