Core Pure: Further Calculus and Series Depth
Further Calculus and Series Depth extends Core Pure calculus and series beyond the basic techniques. In calculus it covers improper integrals (an infinite limit, or a limit where the integrand is undefined at an endpoint), evaluated by replacing the problem bound with a variable and taking a limit; the mean value of a function f(x) on [a,b], given by (1/(b-a)) times the integral of f(x) from a to b; and the area of the curved surface formed when a curve is rotated about the x-axis. In series it covers building the Maclaurin series of a composite function from a standard series by substitution, and using the method of differences on general terms that telescope over more than one step.
Before you start
Make sure you're comfortable with these topics first:
Method
- Recognise an improper integral (an infinite limit, or a limit where the integrand is undefined at an endpoint) and evaluate it by replacing the problem bound with a variable such as R, integrating as normal, then taking the limit as R tends to infinity or to the singular value; the integral converges if this limit is finite.
- Find the mean value of a function f(x) on the interval [a,b] using mean value = (1/(b-a)) * integral from a to b of f(x) dx.
- Find the area of the curved surface formed when y = f(x), for a <= x <= b, is rotated 2*pi radians about the x-axis, using S = integral from a to b of 2*pi*y*sqrt(1+(dy/dx)^2) dx.
- Simplify the expression under the square root, 1+(dy/dx)^2, fully before integrating; it often reduces to a simple linear or quadratic expression in x.
- Build the Maclaurin series of a composite function (such as ln(1+kx) or e^(kx)) by substituting into a standard series (for e^x, sin x, cos x, ln(1+x) or (1+x)^n), then apply the same substitution to the standard series' range of validity to find the new range.
- For the method of differences, write the general term u_r as a difference f(r) - f(r+k) for some fixed k, then list enough of the first and last few terms of the sum to see exactly which terms cancel, especially when k > 1 so more than two boundary terms survive.
- Combine the standard results for sum of r, sum of r^2 and sum of r^3 to evaluate a sum with a polynomial general term, factorising the final answer fully.
Worked example
The curve y = sqrt(x), for 0 <= x <= 3, is rotated through 2*pi radians about the x-axis. Find the exact area of the curved surface generated.
- Differentiate: y = x^(1/2), so dy/dx = (1/2)x^(-1/2), and (dy/dx)^2 = 1/(4x).
- Form 1 + (dy/dx)^2 = 1 + 1/(4x) = (4x+1)/(4x).
- Write the surface area integral: S = integral from 0 to 3 of 2*pi*y*sqrt(1+(dy/dx)^2) dx = integral from 0 to 3 of 2*pi*sqrt(x)*sqrt((4x+1)/(4x)) dx.
- Simplify the integrand: sqrt(x)*sqrt((4x+1)/(4x)) = sqrt(x)*sqrt(4x+1)/(2*sqrt(x)) = sqrt(4x+1)/2, so S = integral from 0 to 3 of pi*sqrt(4x+1) dx.
- Integrate using u = 4x+1: integral of sqrt(4x+1) dx = (1/6)(4x+1)^(3/2) + C.
- Evaluate: S = pi * (1/6)[(13)^(3/2) - (1)^(3/2)] = (pi/6)(13*sqrt(13) - 1).
- Final answer: S = (pi/6)(13*sqrt(13) - 1).
Practice questions
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Q1Evaluate the improper integral from x=1 to infinity of 1/x^3 dx, or state that it diverges.Show answer
Answer: 1/2 (converges); rewrite as the limit as R tends to infinity of [-1/(2x^2)] from 1 to R = 1/2 - 1/(2R^2), which tends to 1/2.
Q2Find the mean value of f(x) = x^2 on the interval [1,4].Show answer
Answer: 7 (mean value = (1/3) * integral from 1 to 4 of x^2 dx = (1/3)*[x^3/3] from 1 to 4 = (1/3)*(63/3) = 7).
Q3Find the first three terms of the Maclaurin series for e^(3x), in ascending powers of x.Show answer
Answer: 1 + 3x + (9/2)x^2 (substitute u=3x into 1+u+u^2/2+...).
Q4Use the method of differences on the identity 1/(r+1) - 1/(r+2) = 1/((r+1)(r+2)) to find sum from r=1 to n of 1/((r+1)(r+2)) as a single fraction in n.Show answer
Answer: n/(2(n+2)) (the sum telescopes to (1/2)[1/2 - 1/(n+2)]).
Q5Find the exact surface area generated when the line y=3x, for 0<=x<=2, is rotated 2*pi about the x-axis.Show answer
Answer: 12*pi*sqrt(10) (dy/dx=3, so 1+(dy/dx)^2=10; S = integral from 0 to 2 of 2*pi*3x*sqrt(10) dx = 6*pi*sqrt(10)*[x^2/2] from 0 to 2).
Q6State the range of values of x for which the Maclaurin series for e^(5x) is valid.Show answer
Answer: All real x (the Maclaurin series for e^x converges for every real x, and substituting 5x for x does not restrict this).
Q7Use the standard results for sum of r and sum of r^2 to find sum from r=1 to n of (4r^2 - 3r), simplifying fully.Show answer
Answer: n(n+1)(8n-5)/6 (= 4*[n(n+1)(2n+1)/6] - 3*[n(n+1)/2], combined over a common denominator and factorised).
Exam-style questions
Written in the style of a A Level Further Maths exam paper, with a full mark scheme.
Show that the integral from x=0 to infinity of x*e^(-2x) dx converges, and find its exact value.
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Given that ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ... for -1 < x <= 1, use this series to find the Maclaurin expansion of ln(1+3x) up to and including the term in x^3, stating the range of values of x for which the expansion is valid.
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Show that (r+1)! - r! = r*r!. Hence use the method of differences to prove that sum from r=1 to n of r*r! = (n+1)! - 1, and evaluate sum from r=1 to 6 of r*r! exactly.
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See real A Level Further Maths past-paper questions, with official mark schemes →
Free printable worksheet
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