1.9 Calculus
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Question 409
The cross-section of a industrial drainage channel is modeled by a curve C C\,C with equation
y=(x−k)2x,x>0 y = \frac{(x - k)^2}{\sqrt{x}}, \quad x > 0 y=x(x−k)2,x>0where k k\,k is a positive constant.
a.[4]
Show that
∫116(x−k)2x dx=ak2+bk+20465 \int_{1}^{16} \frac{(x - k)^2}{\sqrt{x}} \, dx = ak^2 + bk + \frac{2046}{5} ∫116x(x−k)2dx=ak2+bk+52046where a a\,a and b b\,b are integers to be found.
b.[2]
A sketch of the curve C C\,C and a straight line l l\,l are shown. The line l l\,l represents the water level during a flood, intersecting the curve C C\,C at point A(1,9)A(1, 9)A(1,9) and at point B(16,q)B(16, q)B(16,q), where q q\,q is a constant.
Show that k=4k = 4k=4.
c.[4]
The region R R\,R is the cross-sectional area of the water, bounded by the curve C C\,C and the line l l\,l between points A A\,A and BBB. Using the results from parts (a) and (b),
find the area of region RRR.
1.9 Calculus Questions
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.