This comprehensive document delves into the fundamental concepts and principles of calculus, providing a detailed exploration of the subject's core topics. With clear explanations and concise examples, it covers:
- Limits and continuity
- Derivatives and differentiation rules
- Applications of...
Objectives:
✠ Introduces the reverse process of differentiation, which is
called ’integration’
✠ Explains what is meant by an ’indefinite integral’ and a
’definite integral’
✠ Shows how integration can be used to find the area under
a curve
, Overview of Integral Culculus
✠ In some applications it is necessary to find
the area under a curve and above the x
axis between two values of x, say a and b.
✠ Such an area is shown shaded in Figure 1
✠ In this chapter we shall show how this area
can be found using a technique known as
integration
✠ This area of study is known as integral
calculus.
✠ However, before we do this it is necessary
to consider two types of integration: Figure: The shaded area is
indefinite and definite integration. found by integration
, Indefinite integration: the reverse of differentiation
✠ Sometimes we are interested in knowing where certain
things come from.
✠ Suppose we ask ’what function can we differentiate to give
2x?’
✠ In fact, there are lots of correct answers to this question.
✠ We can use our knowledge of differentiation to find these.
dy
✠ The function y = x2 has derivative dx = 2x, so clearly x2
is one of the answer.
✠ Furthermore, y = x2 + 13 also has derivative 2x, as do
y = x2 + 5 and y = x2 + 7, because differentiation of any
constant equals zero.
✠ Indeed any function of the form y = x2 + c where c is a
constant has derivative equal to 2x.
✠ This is differentiation in reverse. We call this process
integration or more precisely indefinite integration.
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