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.
438 exam-style questions on OCR A Level Maths 1.8 Integration, covering 1.8.1 Fundamental theorem of calculus (A-level only), 1.8.2 Integrating x^n, 1.8.3 Integrating standard functions (A-level only), 1.8.4 Evaluating definite integrals, 1.8.5 Area between a curve and the x-axis, 1.8.6 Area between two curves, 1.8.7 Integration as the limit of a sum (A-level only), 1.8.8 Integration by substitution (A-level only), 1.8.9 Integration by parts (A-level only), 1.8.10 Use of partial fractions in integration (A-level only), 1.8.11 Differential equations with separable variables (A-level only), 1.8.12 Interpreting the solution of a differential equation (A-level only), and 1.8 Integration. Each one has a worked solution and a mark scheme showing where the marks go.