Question 1008b

Definite Integrals: Areas and Volumes

2008 Paper 2 Question 2 Variant

Question

The curve

{C}

has equation

y^2 = x \sqrt{1 - x}.

{C}

has

{x\textrm{-intercepts}}

{0}

and

{1.}

The region enclosed by

{C}

, both above and below the

{x\textrm{-axis}}

, is denoted by

{R.}

(i)

Write down an integral that gives the area of

{R,}

and evaluate this integral numerically.

[3]

(ii)

The part of

{R}

above the

{x\textrm{-axis}}

is rotated

{2\pi}

radians about the

{x\textrm{-axis}}

. By using the substitution

{u=1 - x,}

or otherwise, find the exact value of the volume obtained.

[3]

(iii)

Find the exact

{x\textrm{-coordinate}}

of the maximum point of

{C.}

[3]