Definite integrals

Question code: 32

Question 0905

The region

{R}

is bounded by the curve

{y=4 x^2 \mathrm{e}^{ 2 x }}

, the line

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

and part of the

{x\textrm{-axis}}

between

{0}

and

{\frac{1}{2}}

. Find

(i)

the exact area of

{R,}

(ii)

the numerical value of the volume of revolution formed when

{R}

is rotated completely about the

{x\textrm{-axis,}}

giving your answer correct to 3 decimal places.

Answer

(i)

{\left(\frac{1}{2} \mathrm{e} - 1\right) \textrm{ units}^2.}

(ii)

{1.724\textrm{ units}^3.}

Question code: 32