Parametric Equations• To draw a graph from parametric equations, plot the points on the curve given by different values of the parameter.• Eliminate the parameter to give the cartesian equation of the curve.• Parametric equations of circles: - Circle centre (0,0) and radius r - Circle centre (a,b) and radius r•
Techniques for integrationVolumes of revolution• About the x axis:• About the y axis:• Trapezium rule, with n strips of width h:Note that this gives an overestimation of the area under the curve.
Vectors• Magnitude-direction form: (2 dimensions) measured anticlockwise• Component form: from horizontal• Position vector is from origin to point P• Vector• ‘r’ denotes position vector of a general point
Vector equations• Vector equation of the line through A with direction u is given by:• Vector equation through points A and B is given by:• Equation of line through in direction is given by: Cartesian form Vector form
Angle between two vectors• The angle between a and b is given by• Where in two dimensions in three dimensions
• Cartesian equation of a plane perpendicular to is:• Equation of the plane through the point with position vector a, and perpendicular to n, is given by (r-a).n=0.
Differential equations• A differential equation involves derivatives such as• First-order differential equation involves only a first derivative• Some first-order differential equations can be solved by separating the variables• In a general solution you leave the constant of integration in the solution, and in a particular solution you use additional information to calculate the constant of integration.