Mathematics: INTEGRATION Topical Past Paper Questions

Consider a function $f(x)$, for $x \geq 0$. The derivative of $f$ is given by $f'(x) = \frac{6x}{x^{2} + 4}$. (a) Show that $f''(x) = \frac{24 - 6x^{2}}{(x^{2} + 4)^{2}}$. The graph of $f$ is concave-down when $x > n$. (b) Find the least value of $n$. (c) Find $ \int\frac{6x}{x^{…

[Maximum mark: 16] Let $ f(x)=x^{4}-54x^{2}+60x $, for $ -1\leq x\leq6 $. The following diagram shows the graph of f. There are x-intercepts at x = 0 and at x = p. There is a maximum at point A where x = a, and a point of inflexion at point B where x = b. (a) Find the value of p.…

All lengths in this question are in metres. Consider the function $ f(x) = \sqrt{\frac{4 - x^{2}}{8}} $, for $ -2 \leq x \leq 2 $. In the following diagram, the shaded region is enclosed by the graph of f and the x-axis. diagram not to scale A container can be modelled by rotatin…