Figure 1 shows part of the curve with equation $y = \sqrt{\tan x}$ - Edexcel - A-Level Maths Pure - Question 2 - 2007 - Paper 8

To apply the trapezium rule, we first calculate the area using the formula:

$$\text{Area} = \frac{1}{2} (b_1 + b_2) h$$

First, we calculate the widths:

• $h = \frac{\pi}{16}$
• $b_1 = 0.44600$, $b_2 = 0.64350$, $b_3 = 0.81742$, and we sum the bases:

$$\text{Area} = \frac{\pi}{32} \left(0 + 2 (0.44600 + 0.64350 + 0.81742)\right)$$

Calculating that gives:

• Area = $\approx 0.4726$ rounded to 4 decimal places.

To find the volume of the solid generated by rotating the region $\mathcal{R}$ around the x-axis, we use:

$$V = \int_0^{\frac{\pi}{4}} \pi [\sqrt{\tan x}]^2 dx = \pi \int_0^{\frac{\pi}{4}} \tan x \, dx$$

Calculating this, we have:

$$= \pi [\ln(\sec(\frac{\pi}{4})) - \ln(\sec(0))] = \pi \left(\ln(\sqrt{2}) - 0\right) = \frac{\pi}{2} \ln(2).$$