A researcher is studying the intensity of light propagation through a specific lens assembly. The calculation of the phase shift involves the integral:
I=∫1r2r2−16 dr I = \int \frac{1}{r^2 \sqrt{r^2 - 16}} \, dr I=∫r2r2−161drConsider the variable transformation v=secϕv = \sec \phiv=secϕ.
(i) Express vvv in terms of cosϕ\cos \phicosϕ.
(ii) Hence, show that dvdϕ=secϕtanϕ\frac{dv}{d\phi} = \sec \phi \tan \phidϕdv=secϕtanϕ.
(iii) Prove that for 0<ϕ<π20 < \phi < \frac{\pi}{2}0<ϕ<2π, v2−1v=sinϕ\frac{\sqrt{v^2-1}}{v} = \sin \phivv2−1=sinϕ.
(i) Use the substitution r=4secϕr = 4 \sec \phir=4secϕ to show that for r>4r > 4r>4, the integral III can be expressed as:
I=k∫cosϕ dϕ I = k \int \cos \phi \, d\phi I=k∫cosϕdϕwhere kkk is a constant to be found.
(ii) Hence, show that
∫1r2r2−16 dr=r2−1616r+C \int \frac{1}{r^2 \sqrt{r^2 - 16}} \, dr = \frac{\sqrt{r^2 - 16}}{16r} + C ∫r2r2−161dr=16rr2−16+C825 exam-style questions on OCR (MEI) A Level Maths 1.9 Calculus, covering 1.9.1 Gradient of a curve at a point, 1.9.2 Gradient as limit of chord gradient, 1.9.3 Derivative as gradient of tangent, 1.9.4 Sketch the gradient function, 1.9.5 Differentiate y = kx^n, 1.9.6 Second derivative as rate of change of gradient, 1.9.7 Stationary points: maxima and minima, 1.9.8 Increasing and decreasing functions, 1.9.9 Tangent and normal at a point, 1.9.10 Differentiate e^kx, a^kx and ln x (A-level only), 1.9.11 Differentiate trigonometric functions (A-level only), 1.9.12 Product rule (A-level only), 1.9.13 Quotient rule (A-level only), 1.9.14 Chain rule (A-level only), 1.9.15 Rates of change with the chain rule (A-level only), 1.9.16 Implicit differentiation (A-level only), 1.9.17 Concavity and the second derivative (A-level only), 1.9.18 Points of inflection (A-level only), 1.9.19 Integration as reverse of differentiation, 1.9.20 Integrate kx^n, 1.9.21 Constant of integration, 1.9.22 Indefinite and definite integrals, 1.9.23 Area between a graph and the x-axis, 1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only), 1.9.25 Integration as the limit of a sum (A-level only), 1.9.26 Area between two curves (A-level only), 1.9.27 Integration by substitution (reverse chain rule) (A-level only), 1.9.28 Integration by substitution (other cases) (A-level only), 1.9.29 Integration by parts (A-level only), 1.9.30 Integration using partial fractions (A-level only), 1.9.31 Formulate first order differential equations (A-level only), 1.9.32 Solve first order differential equations (A-level only), and 1.9.33 Interpret solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.