C2 Chapter 11 Integration Dr J Frost Last modified: 17 th October 2013

Recap ? ? ? ? ?

Definite Integration We could add together the area of individual strips, which we want to make as thin as possible…

Definite Integration

 Reflecting on above, do you think the following definite integrals would be positive or negative or 0?  + 0   + 0   +0

Evaluating Definite Integrals We use square brackets to say that we’ve integrated the function, but we’re yet to involve the limits 1 and 2. Then we find the difference when we sub in our limits. ? ? ?

Evaluating Definite Integrals ? ? Bro Tip: Be careful with your negatives, and use bracketing to avoid errors.

Exercise 11B a c e ? ? ? ? ? ?

Harder Examples Sketch: (Hint: factorise!) ? ? ?

Harder Examples -3 1 The SketchThe number crunching ??

Exercise 11C ? ? ? ? ?

Curves bound between two lines

How could we use a similar principle if we were looking for the area bound between two lines? ? therefore area… ?

Curves bound between two lines Bro Tip: We’ll need to find the points at which they intersect. ?

Curves bound between two lines Edexcel C2 May 2013 (Retracted) ? ?

y = x(x-3) y = 2x AB C More complex areas Bro Tip: Sometimes we can subtract areas from others. e.g. Here we could start with the area of the triangle OBC. ?

Exercise 11D ? ? ? ?

(Probably more difficult than you’d see in an exam paper, but you never know…) Q6 7 7 ?

y1y1 y2y2 y3y3 y4y4 hhh Trapezium Rule Instead of infinitely thin rectangular strips, we might use trapeziums to approximate the area under the curve. What is the area here? ?

Trapezium Rule In general: width of each trapezium Area under curve is approximately x y ? Example ?

? ? Trapezium Rule May 2013 (Retracted)

To add: When do we underestimate and overestimate?