Revision for integration as the reverse of differentiation, standard integrals, definite integrals, areas and (4PM1) volumes of revolution. Both boards. Cambridge 0606 integrates 1/x and 1/(ax + b); Edexcel 4PM1 does not, but asks for volumes of revolution about the axes.
A definite integral gives the area between a curve and the x-axis when the curve is above the axis.
Area between two curves = ∫(upper − lower) dx between the x-coordinates where they meet.
Volume when y = f(x) is rotated 360° about the x-axis: V = π∫y² dx.
Given dy/dx and a point, integrate and use the point to find c.
Watch out: If a region lies below the x-axis the integral is negative. Work out each part separately and add the sizes.
Worked example· 2 marks
Find \(\displaystyle\int (6x^2 - 4x + 3)\,dx\).
M1 powers increased by one and divided
A1 2x³ − 2x² + 3x + c
Integrate term by term: ∫xⁿ dx = xⁿ⁺¹/(n + 1).
6x³/3 − 4x²/2 + 3x + c = 2x³ − 2x² + 3x + c.
Answer: \(2x^3 - 2x^2 + 3x + c\)
Integration questions
Original exam-style questions, written for this site and each re-solved independently. Tags show the courses whose content a question tests and whether it can be done without a calculator (Cambridge 0606 Paper 1 is non-calculator).
Find the area of the region enclosed by the curve \(y = x(4 - x)\) and the line \(y = x\).
Show the mark scheme and worked solution
M1 x(4 − x) = x gives x = 0, 3
M1 ∫(3x − x²) dx from 0 to 3
A1 9/2
4x − x² = x, so x² − 3x = 0 and x = 0 or x = 3.
Area = ∫ from 0 to 3 of (4x − x² − x) dx = ∫(3x − x²) dx.
[3x²/2 − x³/3] from 0 to 3 = 27/2 − 9 = 9/2.
Answer: \(\frac{9}{2}\)
Question 4 · 2 marks4PM1no calculator
The region bounded by the curve \(y = \sqrt{x}\), the \(x\)-axis and the line \(x = 4\) is rotated through \(360^\circ\) about the \(x\)-axis. Find the exact volume of the solid formed.
Show the mark scheme and worked solution
M1 V = π∫y² dx = π∫x dx
A1 8π
V = π ∫ from 0 to 4 of y² dx = π ∫ from 0 to 4 of x dx.
= π [x²/2] from 0 to 4 = π × 8 = 8π.
Answer: \(8\pi\)
9 more integration questions
Each with a mark scheme and a worked solution. Included with IGCSE plans, free trials and school licences.
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