Equations and inequalities

Question code: 4

Question 0104

A curve has equation

y = \mathrm{e}^{x} - 4 x.

Find the exact volume obtained when the region bounded by the curve, the

{x}

-axis and the lines

{x=4}

and

{x=5}

is rotated

{2 \pi}

radians about the

{x}

-axis.

The two roots of the equation

{\mathrm{e}^{x} - 4 x = 0}

are denoted by

{\alpha}

and

{\beta,}

where

{\alpha < \beta.}

Find the values of

{\alpha}

and

{\beta,}

each correct to 3 decimal places.

Solve

{\mathrm{e}^{x} - 4 x < 0.}

Find the area bounded by the curve and the

{x}

-axis, giving your answer correct to 2 decimal places.

Answer

\pi \left( \frac{1}{2} \mathrm{e}^{10} - \frac{1}{2} \mathrm{e}^{8} - 32 \mathrm{e}^{5} + 24 \mathrm{e}^{4} + \frac{976}{3} \right) \allowbreak \textrm{ units}^3.

{\alpha = 0.357, \, \beta = 2.153.}

{0.357 < x < 2.153.}

{1.83\textrm{ units}^2}

Question code: 4