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1.9 Calculus

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Question 151

The profile of a specialized aerodynamic sail is modeled by a curve with parametric equations

x=36−6t,y=t336+6t,0≤t≤6 x = \sqrt{36-6t}, \quad y = \frac{t^3}{\sqrt{36+6t}}, \quad 0 \le t \le 6 x=36−6t​,y=36+6t​t3​,0≤t≤6

The curve intersects the yyy-axis at the point where t=6t=6t=6 and the xxx-axis at the point where t=0t=0t=0. The region RRR is bounded by the curve and the positive xxx and yyy axes.

a.

Show that the area of RRR is given by

K∫06t31296−36t2 dt K \int_{0}^{6} \frac{t^3}{\sqrt{1296-36t^2}} \, dt K∫06​1296−36t2​t3​dt

where KKK is a constant to be found.

[4]
b.

Using the substitution u=1296−36t2u = 1296 - 36t^2u=1296−36t2, or otherwise, find the exact area of RRR.

[7]
Markscheme

1.9 Calculus Questions

  1. A Level
  2. /Maths
  3. /1.9 Calculus

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.

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