Question 1010

Definite Integrals: Areas and Volumes

2010 Paper 1 Question 6 Variant

Question

Consider the curve with equation

y=x^3 - \frac{4}{3} x + \frac{1}{2}

and the line with equation

{y=\frac{1}{2}.}

The curve crosses the

{x\textrm{-axis}}

{x=\alpha,}

{x=\beta}

and

{x=\gamma}

and has turning points at

{x=- \frac{2}{3}}

and

{x=\frac{2}{3}.}

It is also given that

{\alpha < \beta < \gamma.}

(i)

Find the values of

{\beta}

and

{\gamma,}

giving your answers correct to 3 decimal places.

[2]

(ii)

Find the area of the region bounded by the curve and the

{x\textrm{-axis}}

between

{x=\beta}

and

{x=\gamma.}

[2]

(iii)

Use a non-calculator method to find the area of the region between the curve and the line that is above the line.

[4]

(iv)

Find the set of values of

{k}

for which the equation

x^3 - \frac{4}{3} x + \frac{1}{2}=k

has three real distinct roots.

[2]