Question 1014

Definite Integrals: Areas and Volumes

2014 Paper 1 Question 7 Variant

Question

It is given that

f(x) = x^6 - 3 x^4 - 2.

The curve with equation

{y=f(x)}

crosses the positive

{x\textrm{-axis}}

{x=\alpha,}

and the curve and the line

{y=-2}

meet where

{x=0}

and

{x=\beta.}

(i)

Find the value of

{\alpha,}

giving your answer correct to 3 decimal places, and find the exact value of

{\beta.}

[2]

(ii)

Evaluate

\int_\beta^\alpha f(x) \; \mathrm{d}x,

giving your answer correct to 3 decimal places.

[2]

(iii)

Find, in terms of

{\sqrt{3},}

the area of the finite region bounded by the curve and the line, for

{x\geq 0.}

[3]

(iv)

Show that

{f(x)=f(-x).}

What can be said about the six roots of the equation ${f(x)=0.}$

[4]