Further calculus

What further calculus covers

Integration promoted: areas spun into solids, integrals that run to infinity and still converge, and the inverse trigonometric functions joining the derivative shelf. Core Pure content, examinable on both compulsory papers of 9FM0. Substitution and integration by parts from A level Mathematics are both assumed throughout.

The main ideas

• Volumes of revolution about the x-axis and about the y-axis, with the ordinate squared before anything is integrated.
• Solids generated between two curves, by subtracting the two volumes, and volumes from parametric equations.
• The mean value of a function on an interval, as the integral divided by the width of the interval.
• Improper integrals with an infinite limit or an undefined endpoint, evaluated as a limit and given an explicit verdict.
• Singularities inside the interval, which split the integral and need both halves to converge.
• Differentiating arcsin, arccos and arctan by implicit differentiation.
• Integrating to arcsin and arctan, reached by completing the square, by partial fractions with a quadratic factor, or by a trigonometric substitution.

The results it turns on

V = π∫y² dx about the x-axis

with x² dy and y-limits about the y-axis instead

mean value = (1/(b − a))∫f(x) dx from a to b

the level line enclosing the same area

integrate up to t, then take the limit as t → ∞

the routine an improper integral is written with

∫1/√(a² − x²) dx = arcsin(x/a)

the arcsin pattern, which carries no extra factor

∫1/(a² + x²) dx = (1/a) arctan(x/a)

the arctan pattern, which does carry one

x = a sin θ collapses a² − x², x = a tan θ collapses a² + x²

the substitutions for forms beyond the two patterns

Where it usually goes wrong

• The ordinate is squared first and integrated second. Integrating y and squaring afterwards is the classic error in the volumes lesson.
• For a region between two curves, subtract the two solids. Squaring the gap between the curves gives a different and wrong answer.
• Integrating straight across a singularity inside the interval can return a finite number that is simply wrong: 1/x² across the interval from −1 to 1 gives −2 that way.
• The arctan pattern carries a factor of 1/a and the arcsin pattern does not, and a is identified from what sits with the constant, so 9 + x² has a = 3.

Where to start

Volumes of revolution first, since it is the most mechanical of the three. Mean values and improper integrals second, and practise stating the verdict in words. Inverse trigonometric calculus last, and pair it with the hyperbolic integrals, which sort by the sign inside the root.