Given that y=tanxy = \tan xy=tanx, use the quotient rule to show that
dydx=sec2x \frac{dy}{dx} = \sec^2 x dxdy=sec2xThe cross-section of a high-precision optical lens is modeled by the region bounded by the curve y=3tan2x+2y = 3\tan^2 x + 2y=3tan2x+2, the xxx-axis, and the vertical boundaries x=π6x = \frac{\pi}{6}x=6π and x=π3x = \frac{\pi}{3}x=3π.
Show that the area of this cross-section is
33−3−π6 3\sqrt{3} - \sqrt{3} - \frac{\pi}{6} 33−3−6πwhich simplifies to
23−π6 2\sqrt{3} - \frac{\pi}{6} 23−6πFully justify your answer.
480 exam-style questions on AQA A Level Maths 1.11 H: Integration, covering 1.11.1 Fundamental Theorem of Calculus, 1.11.2 Integrating standard functions, 1.11.3 Definite integrals and areas, 1.11.4 Integration as the limit of a sum (A-level only), 1.11.5 Integration by substitution and by parts (A-level only), 1.11.6 Integration using partial fractions (A-level only), 1.11.7 Differential equations with separable variables (A-level only), and 1.11.8 Interpreting solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.