The diagram shows a sketch of part of the curves with equations $y = 2 x^{3} + 2 x^{2} - 9 x - 24$ and $y = a x^{3} + b x^{2} + c x + d,$ where a, b, c and d are constants with $a \neq 0$ .

q8a-5-4-further-integration-veryhard-ib-aa-sl

The x-coordinates of the points of intersection of the two curves are p, q and r, where $p < q < r .$ Region S is the region enclosed by the two curves between $x = p$ and $x = q$ , while region T is the region enclosed by the two curves between $x = q$ and $x = r .$

The diagram below shows a sketch of part of the curve with equation $y = 4 x^{3} + 9 x^{2} - 16 x - 21 .$

7_LCsAWI_q9-5-4-further-integration-hard-ib-aa-sl

The curve intersects the x-axis at the points $(p, 0), (q, 0)$ and $(r, 0)$ , and region U is the region enclosed by the curve and the x-axis between $x = p$ and $x = q$ .

Given that the areas of regions S and U are equal, calculate the total area enclosed by the two curves in the first diagram. Be sure to provide a suitable justification for your answer.

Further Integration (DP IB Maths: AA HL)

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