Find the area of the region enclosed by the curves \(y = {x^3}\) and \(x = {y^2} - 3\) .

intersection points     A1A1

Note: Only either the x-coordinate or the y-coordinate is needed.

EITHER

\(x = {y^2} - 3 \Rightarrow y = \pm \sqrt {x + 3} \,\,\,\,\,\left( {{\text{accept }}y = \sqrt {x + 3} } \right)\)     (M1)

\(A = \int_{ - 3}^{ - 1.111...} {2\sqrt {x + 3} \,{\text{d}}x + \int_{ - 1.111...}^{1.2739...} {\sqrt {x + 3}  - {x^3}{\text{d}}x} } \)     (M1)A1A1

= 3.4595... + 3.8841...

= 7.34 (3sf)     A1

OR

\(y = {x^3} \Rightarrow x = \sqrt[3]{y}\)     (M1)

\(A = \int_{ - 1.374...}^{2.067...} {\sqrt[3]{y}}  - ({y^2} - 3){\text{d}}y\)     (M1)A1

= 7.34 (3sf)     A2

[7 marks]

This question proved challenging to most candidates. Just a few candidates were able to calculate the exact area between curves. Those candidates who tried to express the functions in terms x of instead of y showed better performances. Determining only \(\sqrt {x + 3} \) was a common error and forming appropriate definite integrals above and below the x-axis proved difficult. Although many candidates attempted to sketch the graphs, many found only one branch of the parabola and only one point of intersection; as the graph of the parabola was not complete, many candidates did not know which area they were trying to find. Not many split the integral correctly to find areas that would add up to the result. Premature rounding was usually seen and consequently final answers proved inaccurate.