INTEGRATION Part #1 – Functions Integration

Objective After this topic, students will be able to ddefine and apply the suitable method to solve indefinite and definite integration

History of integration  Archimedes is the founder of surface areas and volumes of solids such as the sphere and the cone.  His integration method was very modern since he did not have algebra, or the decimal representation of numbers  Gauss was the first to make graphs of integrals, and with others continued to apply integrals in the mathematical and physical sciences.  Leibniz and Newton discovered calculus and found that differentiation and integration undo each other

The Use of Integration Integration was used to design the Petronas Towers making it stronger Many differential equations were used in the designing of the Sydney Opera House Finding areas under curved surfaces, Centers of mass, displacement and Velocity, and fluid flow are other uses of integration Finding the volume of wine casks. Was one of the first uses of integration.

Reverse the differentiation process 5 Function Derivative Differentiate What could we call the reverse process? ?

6 What is the opposite to “social “behaviour? Anti-social behaviour

Reverse the differentiation process 7 Function Derivative Differentiate Anti- differentiate

Antiderivatives 1.Lecturer will give you Students’ Worksheet #1 2.Work in groups

Call Back 1.I n the example you just did, we know that F(x) = x2 is not the only function whose derivative is f(x) = 2x G(x) = x2 + 2 has 2x as the derivative H(x) = x2 – 7 has 2x as the derivative 2.F or any real number, C, the function F(x) = x2 + C has f(x) as an antiderivative

conclusions TThere is a whole family of functions having 2x as an antiderivative TThis family differs only by a constant

Antiderivatives Since the functions G(x) = x 2 F(x) = x H(x) = x 2 – 7 differ only by a constant, the slope of the tangent line remains the same The family of antiderivatives can be represented by F(x) + C

Indefinite Integral The family of all antiderivatives of f is indicated by Integral sign Integrand This is called the indefinite integral and is the most general antiderivative of f The name of variable involved

Formula

Why we include C  The derivative of a constant is 0  However, when you integrate, you should consider that there is a possible constant involved, but we don’t know what it is for a particular problem  Therefore, you can just use C to represent the value.  To solve for C, you will be given a problem that gives you the y(0) value.  Then you can plug the 0 in for x and the y(0) value for y.

Example

Rules of Integration

e xample

Extension Rule

Example

Integration of Exponential Functions

Example

Example