This block contains Units 25 - 28Unit 25 Multiple IntegralsThis unit generalises the idea of an integral still further to deal with two and three dimensions by introducing two new kinds of integrals, called area integrals and volume integrals. How area integrals can be evaluated as combinations of two ordinary integrals, is shown and applications of area integrals, including the evaluation of centres of mass of planar (i.e. two-dimensional) objects are described. How volume integrals can be expressed as combinations of three ordinary integrals and how area integrals can be used to compute the area of a curved surface are also demonstrated.Unit 26 Numerical methods of Differential EquationsThis unit introduces the study of numerical methods for differential equations. It covers the Taylor`s theorem with exercises, recaps Euler`s method for solving initial-value problems involving first-order differential equations and goes on to explain that more efficient methods exist. Three new methods known as Runge-Kutta methods, are derived and a way of determining how small the step size h would need to be in order to achieve a given accuracy for a given initial-value problem is established.Unit 27 Rotating Bodies and Angular MomentumThis unit deals with the motion of extended bodies, and in particular with their rotational motion. Rotating bodies, Angular momentum, Rigid-body rotation about a fixed axis and rotation about a moving axis are all covered in this unit.Unit 28 Planetary OrbitsThis unit shows how Newton`s laws of motion and Newton`s law of universal gravitation can be used to predict the orbits of planets around the Sun. In particular, it shows that Kepler`s laws of planetary motion can be derived using Newtonian mechanics.

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