Find the area of the region enclosed between the curves \(f(x) = -x^{2} + 4x + 3\) and \(g(x)\) \(=…

The area is $\frac{49}{12}$. Here $f(x) - g(x)$ $= x^{3} - 8x^{2} + 14x - 2$, which is positive on [1, 2] (it equals 5 at x = 1 and 2 at x = 2, with no root in between). Area $= \int_{1}^{2} (x^{3} - 8x^{2} + 14x - 2)\,dx$ $= \left[\frac{x^{4}}{4} - \frac{8x^{3}}{3} + 7x^{2} - 2x\right]_{1}^{2}$ $= \frac{20}{3} - \frac{31}{12}$ $= \frac{49}{12}$. The other values come from slips in integrating or in subtracting the two curves.