Integral Calculus An antiderivative of f `(x) = f(x) The indefinite integral: You need to remember all the integral identities from higher.

A definite integral is where limits are given. This gives the area under the curve of f `(x) between these limits. Practise can be had by completing page 70 Exercise 1A and 1B TJ exercise 1

Standard forms From the differentiation exercise we know: This gives us three new antiderivatives. Note:

` (We need to use a little trig here and our knowledge of integrals.) From a few pages ago we know Page 72 Exercise 2A and if time some of 2B TJ Exercise 2 and some of 3

Integration by Substitution When differentiating a composite function we made use of the chain rule. and When integrating, we must reduce the function to a standard form – one for which we know the antiderivative. This can be awkward, but under certain conditions, we can use the chain rule in reverse.

If we wish to perform we can proceed as follows. The integral then becomes which it is hoped will be a standard form. Substituting back gives,

Page 74 Exercise 3 Odd Numbers TJ Exercise 4.

For many questions the choice of substitution will not always be obvious. You may even be given the substitution and in that case you must use it.

Substituting gives, We can not integrate this yet. Let us use trig.

Now for some trig play…..

We now need to substitute theta in terms of x.

Page 75 Exercise 4A Odd numbers TJ Exercise 5 and 6 This has been printed………………….

Now for some not very obvious substitutions……………….

Page 76 Exercise 4B Questions 1 to 5

Substitution and definite integrals Assuming the function is continuous over the interval, then exchanging the limits for x by the corresponding limits for u will save you having to substitute back after the integration process.

Page 77 Exercise 5A Odd Questions Plus 1 or 2 from 5B

Special (common) forms Some substitutions are so common that they can be treated as standards and, when their form is established, their integrals can be written down without further ado. Page 80 Exercise 6A Questions 1(c), (d) 10 plus a few more Do some of exercise 6B if time.

Area under a curve a b y = f(x) a b y = f(x)

Area between the curve and y - axis y = f(x)

1. Calculate the area shown in the diagram below. 5 2 y = x2 + 1 This has been Printed. TJ Exercise 7, 8 and 9

Volumes of revolution Volumes of revolution are formed when a curve is rotated about the x or y axis.

Find the volume of revolution obtained between x = 1 and x = 2 when the curve y = x2 + 2 is rotated about (i) the x – axis (ii) the y – axis.

Page 91 Exercise 10B Questions 4, 5 and 10. TJ Exercise 10.

Displacement, velocity, acceleration for rectilinear motion We have already seen in chapter 2 on differential calculus that where distance is given as a function of time, S = f (t), then Hence,

A particle starts from rest and, at time t seconds, the velocity is given by v = 3t2 + 4t – 1. Determine the distance, velocity and acceleration at t = 4 seconds. When t = 0, s = 0 When t = 4,

Page 88 Exercise 10A Questions 1, 2, 4, 6, 7, 8. Do the rest of 10A and some from 10B if time permits. Do the Chapter 3 review exercise on page 93.