2021 ACJC P1 Q8

By using the substitution \(x=\cos \theta \), find \(\displaystyle\displaystyle \displaystyle\int{\frac{{{x}^{3}}}{\sqrt{1-{{x}^{2}}}}\mathrm{d}x}\), expressing your answer in terms of \(x\).[4]

Verify that the curves with equations \(y=\frac{{{x}^{3}}}{\sqrt{1-{{x}^{2}}}}\) and \(y=\frac{1}{\sqrt{49-4{{x}^{2}}}}\) intersect at the point with \(x\)- coordinate \(\frac{1}{2}\).[1]

Hence find the exact area bounded by the curves \(y=\frac{{{x}^{3}}}{\sqrt{1-{{x}^{2}}}}\) and \(y=\frac{1}{\sqrt{49-4{{x}^{2}}}}\), and the \(y\)-axis.[4]