Integration Esme n Esme is approaching her last

Integration

Esme n Esme is approaching her last topic in her Al level mathematics programme. She goes for a walk through the forest to reflect on her two years of study and for her love of mathematics. Esme stumbles across a very special tree.

The Integration Methods Tree Standard Integrals n Inverse of Chain rule n Using Trig Identities n Partial fractions n Substitution n Integration by Parts n

A little later Integration Applications Finding the area using integration and trapezium rule n Finding the volumes of solids of revolution n

Branch 1 Standard Integrals n Match the integral with the correct answers.

Branch 2 Reverse of the Chain Rule Instead of putting the power in front and dropping the power by one and multiplying by the derivative of the bit in the middle n Y = (2 x+4)6 n Y’ = 6(2 x+4)5 X 2 n We raise the power by one, divide by that power and divide by the derivative of the bit in the middle n

The Magic Phrase We raise the power by one, divide by that power and divide by the derivative of the bit in the middle n

Example

Branch 3 Using trig identities to Integrate Do you remember all our trig identities from C 3? n Write down as many as you can remember n

Example 1

More Examples

Example

Branch 4 Using partial fractions n n We can use partial fractions to integrate expressions that are too long to do other methods with! What are those rules again?

Ex 6 D Question 1 e

Examples

Improper Fraction

More Standard Integrals Recognizing

Recognizing the Reverse of the Chain Rule

Recognizing Ln and reverse of Chain rule! n Here remember :

Examples

Example

Another example

Your example

Your example

A volunteer

A volunteer

Branch 5 Integration using Substitution n Here is another post it lesson! Sometimes it is much easier to substitute a simpler function that we can easily integrate Integrate the following using the substitution provided

Example 1 Remember to substitute back and c!

Example 3

Example 3

Definite Integrals Here remember to change your boundaries using the substitution you have used n Let’s look at the first example but this time we want to evaluate this integral from x = 1 to x= 3 n

Example 4 Here you can use the new boundaries 7 and 11! No need to add c or substitute back! Why?

Branch 6 Using Integration by Parts n Do you remember the product rule to differentiate the product of two functions? n Or in words v du + u dv We can actually rearrange this to help us to integrate two functions multiplied together! n

Integration by parts Rearranging Integrating

Integration by parts how do I use this?

Example 2

Example 3 with boundaries

Five review questions

Five review questions

Five review questions

Five review questions

Five review questions

Five review questions

Integrals given in your formulae booklet

Where do they come from?

Trapezium rule revisited n Do you remember this from C 2? Why? n Use a little table as this helps with your working!

Where does this symbol come from? n We know that the integral sign is really the Germanic long "s" as used by Leibniz, and is used to denote the limit of the sum of an increasingly large number of decreasingly sized divisions of an area.

General Areas n So in general the area of a curve y = f(x) which is bound by the lines x = a and x = b and the x axis can be found by

Areas Do you want to see a proof? y δA δx y + δy

Drawing a sketch will help you to determine whether you need to split up your boundary conditions. n Remember any region below the x- axis will give a negative result! n Using you GDC-follow these instructions n

Volumes of solids of revolution n An applet increasing the number of discs

Volumes of solids of revolution

Volumes of solids of Revolution

Example 1 n Find the area and volume of the solid generated by the curve:

Example 1 n Find the area and volume of the solid generated by the curve:

Example 2 n n n The region R is bound by the curve C, the x axis and the lines x=-8 and x=8. The parametric equations for C are x=t 3 and y=t 2. Find the area of R Find the volume of solid of revolution formed when R is rotated about the x axis.

Example 3 volumes n n n The curve has parametric equations x=sin t and y=sin 2 t where 0<t< /2. Find the area bound by the curve Find the volume of solid generated about the x axis.

Example 3 volumes n n n The curve has parametric equations x=sin t and y=sin 2 t where 0<t< /2. Find the area bound by the curve Find the volume of solid generated about the x axis.

Example 3 cont

Example 3 cont

Solving differential equations n n n Separating the variables and then integrating. Sometimes you are given a differential equation with functions in terms of x and y. Find the GS of the following DE. Leave your answer y= f(x).

Solving differential equations n n n Separating the variables and then integrating. Sometimes you are given a differential equation with functions in terms of x and y. Find the GS of the following DE. Leave your answer y= f(x).

Example separating the variable. Find the GS of this DE

One more example

Real life applications! n The rate at which Ewoks increase in population at time t is given by d. P/dt=KP, where P is the population, K is a positive constant. In the beginning of time there were 8 Ewoks on earth and at time t=1 they grew to 56. Find the population at t=2.

Mixed exercises n Example