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Integration (Metric)

Learn, or revise, techniques of integration (parts, substitution, partial fractions, reduction formulae) and applications to areas in the plane and volumes of revolution.

Techniques

Integration by parts, by substitution and with the use of partial fractions (A-level topics); improper integrals and reduction formulae (Further Maths topics).

The use of the formula \(\int u\,dv=u\,v-\int v\,du\).

Integration using substitution, or change of variable.

Integration of rational functions by reducing them to partial fractions.

Integrals between \(a\) and \(\infty\); convergence and non-convergence.

Integration of \(I_n=\int f_n(x)\,dx\) by expression \(I_n\) in terms of \(I_{n-1}\) or \(I_{n-2}\).

Applications

Areas between curves, and volumes of revolution. This is an A-level topic, which Science and Engineering courses at Imperial will assume you know.

Area enclosed by the intersecting curves \(y=f(x)\) and \(y=g(x)\).

Volume of revolution about the \(x\)- and \(y\)-axis.