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1.8 Integration

1.8 Integration

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Question 235

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>0

where k k\,k is a positive constant.

a.

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} ∫116​x​(x−k)2​dx=ak2+bk+52046​

where a a\,a and b b\,b are integers to be found.

[4]
b.

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.

Sketch of curve C and straight line l intersecting at A(1, 9) and B(16, q), with region R between them.

[2]
c.

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.

[4]
Markscheme

1.8 Integration Questions

  1. A Level
  2. /Maths
  3. /1.8 Integration

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.

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