Integral Calculus

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Integral Calculus
An antiderivative of f (x) f(x)
The indefinite integral
You need to remember all the integral identities
from higher.
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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
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Standard forms
From the differentiation exercise we know
This gives us three new antiderivatives.
Note
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(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
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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.
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If we wish to perform
we can proceed as follows.
Substituting back gives,
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Page 74 Exercise 3 Odd Numbers TJ Exercise 4.
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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.
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Substituting gives,
We can not integrate this yet. Let us use trig.
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Now for some trig play..
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We now need to substitute theta in terms of x.
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Page 75 Exercise 4A Odd numbers TJ Exercise 5
and 6
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Now for some not very obvious substitutions.
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Page 76 Exercise 4B Questions 1 to 5
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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.

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Page 77 Exercise 5A Odd Questions Plus 1 or 2
from 5B
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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.
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Area under a curve
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Area between the curve and y - axis
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1. Calculate the area shown in the diagram below.

TJ Exercise 7, 8 and 9
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Volumes of revolution
Volumes of revolution are formed when a curve is
rotated about the x or y axis.
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• 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.

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