Applications of integration: RVHS 2023 Promo Q7

(i)

$$\begin{aligned} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} |\tan x|\,\mathrm{d} x &= \int_{-\frac{\pi}{4}}^{0} -\tan x\,\mathrm{d} x + \int_{0}^{\frac{\pi}{4}} \tan x\,\mathrm{d} x\\ &= 2\int_{0}^{\frac{\pi}{4}} \tan x\,\mathrm{d} x\\ &= 2\Big[\ln(\sec x)\Big]_{0}^{\frac{\pi}{4}}\\ &= 2\left(\ln\frac{1}{\cos\frac{\pi}{4}} - \ln\frac{1}{\cos 0}\right)\\ &= 2\left(\ln\sqrt{2} - \ln 1\right)\\ &= \ln 2. \end{aligned}$$

(ii) $$\begin{aligned} \text{Volume} &= \pi\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \left(|\tan x| + 1\right)^2\,\mathrm{d} x\\ &= \pi\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} |\tan x|^2 + 2|\tan x| + 1\,\mathrm{d} x\\ &= \pi\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \left(\tan^2 x + 1\right)\,\mathrm{d} x + \pi\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} 2|\tan x|\,\mathrm{d} x\\ &= \pi\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sec^2 x\,\mathrm{d} x + 2\pi\ln 2\\ &= \pi\Big[\tan x\Big]_{-\frac{\pi}{4}}^{\frac{\pi}{4}} + 2\pi\ln 2\\ &= \pi\left(\tan\frac{\pi}{4} - \tan\left(-\frac{\pi}{4}\right)\right) + 2\pi\ln 2\\ &= 2\pi(1 + \ln 2)\ \text{units}^3. \end{aligned}$$