WAEC Further Mathematics 2013 Paper

Concept:Use the power rule of integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$.For a definite integral, evaluate the antiderivative at the upper limit and subtract its value at the lower limit.Explanation:First, integrate $8x - 4x^2$ term by term between the limits $0$ and $2$.$\int (8x - 4x^2) \, dx = \frac{8x^2}{2} - \frac{4x^3}{3} = 4x^2 - \frac{4x^3}{3}$.So the definite integral is $\left[4x^2 - \frac{4x^3}{3}\right]_{0}^{2}$.Substitute the upper limit, $x=2$:$4(2^2) - \frac{4(2^3)}{3} = 16 - \frac{32}{3}$.Substitute the lower limit, $x=0$:$4(0^2) - \frac{4(0^3)}{3} = 0$.Subtract the lower-limit value from the upper-limit value:$\left(16 - \frac{32}{3}\right) - 0 = \frac{48 - 32}{3} = \frac{16}{3}$.Answer:$\frac{16}{3}$Therefore, the correct option is C.