Figure 3 shows a sketch of part of the curve C with equation y = 3^x The point P lies on C and has coordinates (2, 9) - Edexcel - A-Level Maths Pure - Question 1 - 2014 - Paper 4

The volume V of the solid formed by rotating region R about the x-axis can be calculated using the disk method:

$V = \pi \int_{0}^{x_Q} (3^x)^2 \, dx$

This simplifies to:

$V = \pi \int_{0}^{x_Q} 9^{x} \, dx$

The integral of ( 9^x ) is:

$\int 9^x \, dx = \frac{9^x}{\ln(9)} + C$

Thus, evaluating the definite integral:

$V = \pi \left[ \frac{9^x}{\ln(9)} \right]_{0}^{x_Q}$

Substituting the limits:

$V = \pi \left( \frac{9^{x_Q}}{\ln(9)} - \frac{9^{0}}{\ln(9)} \right)$

Simplifying further:

$V = \pi \left( \frac{9^{2 - \frac{1}{\ln(3)}}}{\ln(9)} - \frac{1}{\ln(9)} \right) = \frac{\pi}{\ln(9)} \left( 9^{2 - \frac{1}{\ln(3)}} - 1 \right)$

Thus, the volume of the solid generated is:

$V = \frac{p}{q}$

where p and q can be determined based on the expression derived.