Find the value of $a$ and the value of $b$.

The region bounded by the graph of $y = \ln (5x + 10)$, the $x$-axis and the lines $x = {\text{e}}$ and $x = 2{\text{e}}$, is rotated through $2\pi$ radians about the $x$-axis. Find the volume generated.

EITHER

$y = \ln (x - a) + b = \ln (5x + 10)$     (M1)

$y = \ln (x - a) + \ln c = \ln (5x + 10)$

$y = \ln \left( {c(x - a)} \right) = \ln (5x + 10)$     (M1)

OR

$y = \ln (5x + 10) = \ln \left( {5(x + 2)} \right)$     (M1)

$y = \ln (5) + \ln (x + 2)$     (M1)

THEN

$a =  - 2,{\text{ }}b = \ln 5$     A1A1

Note:     Accept graphical approaches.

Note:     Accept $a = 2,{\text{ }}b = 1.61$

[4 marks]

$V = \pi {\int_e^{2e} {\left[ {\ln (5x + 10)} \right]} ^2}{\text{d}}x$     (M1)

$= 99.2$     A1

[2 marks]

Total [6 marks]