Question 1011

Definite Integrals: Areas and Volumes

2011 Paper 1 Question 5 Variant

Question

It is given that

{f(x)=1 - x.}

(i)

On separate diagrams, sketch the graphs of

{y=f(|x|)}

and

{y=|f(x)|,}

giving the coordinates of any points where the graphs meet the

{x\textrm{-}}

and

{y\textrm{-}}

axes. You should label the graphs clearly.

[3]

(ii)

State the set of values of

{x}

for which

f(|x|)=|f(x)|.

[1]

(iii)

Find the exact value of the constant

{a}

for which

\int_{- \frac{1}{2}}^{\frac{1}{2}} f(|x|) \; \mathrm{d}x = \int_{\frac{1}{2}}^a |f(x)| \; \mathrm{d}x.

[3]