The Fundamental Theorem of Calculus Integration

An introduction to integration

Thursday 22nd September 2011 Newton Project

How to find the area under the curve

In this presentation we are going to look at how we can find the area under a curve. In this case the area we are looking to find is the area bounded by both the x axis and the y-axis.

We will then consider how integration might help us do this.Using rectangles to estimate the area

Maybe we could divide the area into rectangles? Can we make the approximation better?Insert YouTube Mr Bartons Maths Area under a curve

Use YouTube Mr Bartons Maths Area under a curve

4Is there a better way?Hint .....Area of a trapeziumA trapezium is a quadrilateral that has only one pair of parallel sides. Consider the area of the following trapezium.

Area of a Trapezium = (a+b) x h 2

habDeriving the FormulaArea of a Trapezium: h( a+b)

T1 = h(y0+y1)T2 = h(y1+y2)T3 = h(y2+y3)T4 = h(yn-1 +yn)Whole Area is the additionAll of the Trapeziums:

A= h(y0+y1+y1+y2+y2+y3+ yn-1 +yn)A = h(y0 + 2(y1+y2+y3+yn-1)+ yn)

Now some examples!

Use Loris handout of examples

7Integration

The next part of this presentation explains the concept of integration, and how we can use integration to find the area under a curve instead of using the trapezium rule.Integration

Consider a typical element bounded on the left by the ordinate through a general point P(x,y).The width of the element represents a small increase in the value of x and can be called and so can be called Also, if A represents the area up to the ordinate through P, then the area of the element represents a small increase in the value of x and so can be called A typical strip is approximately a rectangle of height y and width Therefore, for any elementThe required area can now be found by adding the areas of all the strips from x=a to x=b

x=ax=b

P(x,y)9Therefore, for any elementThe required area can now be found by adding the areas of all the strips from x=a to x=b

The notation for the Total Areas is

so

as gets smaller the accuracy of the results increases

Until in the limiting case Total Area =

Integration

can also be written as

As gets smaller

But so Therefore

The boundary values of x defining the total area are x=a and x=b so this is more correctly written as

Integration