Using double angle formulae, show that \(\cos^4\theta=\frac18(\cos4\theta+4\cos2\theta+3)\). The region \(R\) lies in the first quadrant and is bounded by the curve \(y^4=(9-x^2)^3\), the \(x\)-axis and the lines \(x=1.5\) and \(x=3\). \(R\) is rotated about the \(x\)-axis through \(2\pi\) radians. Using the substitution \(x=3\sin\theta\), find the exact volume generated.

Using double angle formulae, show that \(\cos^4\theta=\frac18(\cos4\theta+4\cos2\theta+3)\).

The region \(R\) lies in the first quadrant and is bounded by the curve \(y^4=(9-x^2)^3\), the \(x\)-axis and the lines \(x=1.5\) and \(x=3\). \(R\) is rotated about the \(x\)-axis through \(2\pi\) radians. Using the substitution \(x=3\sin\theta\), find the exact volume generated.

N2023 P1 Q6

N2023 P1 Q6