Differentiation From First Principle

Summary 4 steps to work out differentiation from the First Principle: Give increments to both x & y i.e . Find change of y. Find rate of change of y with respect to x i.e    or  . Take the limit of    as . Gradient of a curve We know that the gradient of … Read more

Integration By Parts

Summary Remember the formula for ‘Integration By Parts’: We now know how to evaluate many basic integrals. However there are many integrals which are in the form of two functions and cannot be simplified by any substitution. In such cases we use the ‘Product Rule’ of differentiation. We know that: Rewrite it as: Integrating both … Read more

Volume of Solids of Revolution

Summary Rotating about the x axis, Volume of Revolution: Rotating about the y axis, Volume of Revolution: Definite Integrals Formula  We know that definite integrals can help us figure out areas underneath the curves. Recalling the formula for definite integrals from the article ”Integration” : Similarly, pretty much using the same principle we can work … Read more

Area under a curve

Summary Area under a curve can only be calculated if the integral is definite. It must have limits. We must be aware of the three common scenarios when working out the areas under the curves: Area Under the Curve Find area between a curve, the x axis and the line x = a and x … Read more

Differentiation of Trigonometric Functions

Summary Remember these derivatives of the trigonometric functions: Differentiation of sin(x) and cos(x) To begin with, we know that differentiation is a method to find the gradient of a curve. Rule of differentiation is if , then  . However in this article we will focus entirely on differentiation of trigonometric functions. Consider the graph of … Read more

Trapezium Rule

Summary Trapezium rule can only be applied to Definite Integrals. Area under the curve is divided into equally spaced intervals forming a trapezium. Trapezium Rule only provides an estimate area of under the curve. The formula given is: where Definite Integrals Let’s start by briefly recalling the definite integrals from the article ”Integration”. Definite Integration … Read more