Question 1007

Definite Integrals: Areas and Volumes

2007 Paper 2 Question 4 Variant

Question

(i)

Find the exact value of

\int_{\frac{4}{3} \pi}^{\frac{23}{12} \pi} \sin^2 x \; \mathrm{d}x.

Hence find the exact value of

\int_{\frac{4}{3} \pi}^{\frac{23}{12} \pi} \cos^2 x \; \mathrm{d}x.

[6]

(ii)

The region

{R}

is bounded by the curve

y=x^2 \sin x ,

the line

{x=\frac{1}{2} \pi}

and part of the

{x\textrm{-axis}}

between

{0}

and

{\frac{1}{2} \pi.}

Find

(iia)

the exact area of

{R,}

[5]

(iib)

the numerical value of the volume of revolution when

{R}

is rotated completely about the

{x\textrm{-axis},}

giving your answer correct to 3 decimal places.

[2]