An engineer is designing a precision acoustic resonator. The internal radius of the resonator, RRR cm, at a distance xxx cm from the origin is modeled by the function
R(x)=6xe−13x0≤x≤6 R(x) = 6x e^{-\frac{1}{3}x} \quad 0 \le x \le 6 R(x)=6xe−31x0≤x≤6A cross-section of the resonator's interior, denoted by the region SSS, is bounded by the curve, the xxx-axis, and the line with equation x=6x = 6x=6.
The solid of revolution for the resonator's internal chamber is formed by rotating the region SSS through 2π2\pi2π radians about the xxx-axis.
Show that the internal volume, VVV, of this chamber is given by
V=k∫06x2e−23x dx V = k \int_{0}^{6} x^2 e^{-\frac{2}{3}x} \, dx V=k∫06x2e−32xdxwhere kkk is a constant to be determined.
Find ∫x2e−23x dx\int x^2 e^{-\frac{2}{3}x} \, dx∫x2e−32xdx.
A complete resonator is constructed by joining two of these chambers end-to-end at their widest faces. The resulting device has a mass of 0.8 kg and a total length of 12 cm.
Given that density=massvolume\text{density} = \frac{\text{mass}}{\text{volume}}density=volumemass,
find the density of this resonator. Give your answer in g/cm3\text{g/cm}^3g/cm3 to 3 significant figures.
825 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.