The diagram displays the curves given by $y = \sqrt[3]{5x^2 + 7}$ and $y = \frac{27}{2x + 5}$ for $x \geq 0$. The curves intersect at $(2, 3)$. Region $A$ is enclosed by the curve $y = \sqrt[3]{5x^2 + 7}$ together with the straight lines $x = 0$, $x = 2$ and $y = 0$. Region $B$ is enclosed by the two curves and the straight line $x = 0$.
(a)[3]
Apply the trapezium rule using two intervals to estimate the area of region $A$. Give your answer correct to $3$ significant figures.
(b)[3]
Find the exact total area of regions $A$ and $B$. Give your answer in the form $k \ln m$, where $k$ and $m$ are constants.
(c)[1]
Deduce an approximation to the area of region $B$. Give your answer correct to 3 significant figures.
(d)[2]
State, with a reason, whether your answer to part (c) is an over-estimate or an under-estimate of the area of region $B$.
Worked solution & mark scheme
This 9-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Use $y$-values $\sqrt[3]{7},\sqrt[3]{12},\sqrt[3]{27}$ or decimal equivalents” …