Any ‘normal’ continuous functioncan be expressed as a polynomial. A ‘normal’ function has a smooth curve with no corners, so that we can differentiate it as many times as we need to with no restrictions. To say that the function is continuous on an interval means that the graph of the function may be drawn […]

## Integration in Polar Coordinates

For an area which may be approximated as part of a circle we may write Then the area under a curve expressed in polar coordinates may be expressed as For example to find the area under the curve r=1+cos 2 %theta , shown below. To integraterearrangeto get

## Inequalities Involving Integrals and Series

We can estimate the value of the sum of a series of termsby representing the sum by a sequence of rectangles of base 1 and heightforThis can also be estimated using the integral ThenThe integral is an underestimate for the sum of the series since since – we can add a sequence of these expressions […]

## The Parabola

A parabola is the locus of points defined by the condition that the the distance from a point on the curve to a fixed point F is equal to the least distance from that point on the line to a fixed line L: The point F is called the focus. The point L is called […]

## Surfaces of Revolution

From the diagramand the shaded area We can integrate:This equation is for a surface of revolution about the x – axis. For a surface of revolution about the y – axis the equation is Example: Find the Surface of revolution generated when y^2 =4x is rotated about the x – axis.

## Solving Hyperbolic Trigonometric Equations (1)

The simplest hyperbolic trigonometric equations involve one function only which may be sinh, cosh or tanh. For example: Slightly more complex equations involve two functions. We may be able to obtain one function from this hence solve the equation. Divide byto obtainthen divide by 4 to obtain We may have a quadratic hyperbolic equation. We […]

## Expressingin the Formor

The identities(1) and(2) will be useful here. To expressin the form equate both expressions and expand the right hand side using identity (1) Equate coefficients of (1) and coefficients of (2) (1) divided by (2) gives andgives To expressin the form equate both expressions and expand the right hand side using identity (1) Equate coefficients […]

## Eccentricity

The eccentricity of a conic section is given the label For a circle For an ellipseandifandif and For a parabola, For a hyperbolaand and

## Calculation of Curvature – Examples

Calculation of curvature is given using the formula(1) wherefor a curve expressed in cartesian coordinatesand (2) for a curve given in parametric coordinates Example:Find the curvature of Use (1). Now useto obtain Sub into (1). Example:Find the curvature of Use (1). . Sub into (1). Example:Find the curvature of the curve given by the parametric […]

## Proving Hyperbolic Double Angle Trigonometric Formulae

The ordinary double angle formulae of trigonometry have analogous results for the hyperbolic trigonometric functions: Trigonometric Formulae Hyperbolic Trigonometric Formulae The Formulae can be proved using the definitionsand