2007 H2 Mathematics

Paper 2 Question 4

(a)

Find the exact value of $\int_{0}^{\frac{5}{3} π} \sin^{2} x d x .$

Hence find the exact value of $\int_{0}^{\frac{5}{3} π} \cos^{2} x d x .$

[6]

(b)

The region $R$ is bounded by the curve $y = x^{2} \sin x,$ the line $x = \frac{1}{2} π$ and part of the $x -axis$ between $0$ and $\frac{1}{2} π .$ Find

(i)

the exact area of $R,$

[5]

(ii)

the numerical value of the volume of revolution formed when $R$ is rotated completely about the $x -axis,$ giving your answer correct to $3$ decimal places.

[2]