Let and where

MATHEMATICS · Subjective · A Level · Mathematics · Trigonometry · Trigonometry

Problem

Let

f(\theta) = \int_{0}^{1} (x + \sin \theta)^2 dx

and

g(\theta) = \int_{0}^{1} (x + \cos \theta)^2 dx

where

\theta \in [0, 2\pi]

. The quantity

f(\theta) - g(\theta)

\forall \theta

in the interval given in column I, is

Column I	Column II
(A) $\left(\frac{\pi}{4}, \frac{3\pi}{4}\right)$	(p) negative
(B) $\left[\frac{3\pi}{4}, \pi\right]$	(q) positive
(C) $\left[\frac{3\pi}{2}, \frac{7\pi}{4}\right)$	(r) non-negative
(D) $\left(0, \frac{\pi}{4}\right) \cup \left(\frac{7\pi}{4}, 2\pi\right)$	(s) non-positive