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CHAPTER 6 INTEGRATION LEARNING OBJECTIVES Upon completion of this chapter, you should be able to do
the following: 1. Define integration. 2. Find the area under a curve and interpret indefinite
integrals. 3. Apply rules of integration. 4. Apply definite integrals to problem solving. INTRODUCTION The two main branches of calculus are differential
calculus and integral calculus. Having studied differential calculus in
previous chapters, we now turn our attention to integral calculus. Basically,
integration is the inverse of differentiation just as division is the inverse
of multiplication, and as subtraction is the inverse of addition. DEFINITIONS Integration
is defined as the inverse of differentiation.
When we were dealing with differentiation, we were given a function F(x) and
were required to find the derivative of this function. In integration we will
be given the derivative of a function and will be required to find the
function. That is, when we are given the function f(x), we will find another
function, F(x), such that
In
other words, when we have the function f(x), we must find the function F(x) whose derivative is the function f(x). If
we change equation (6.1) to read
we
have used dx as a differential. An equivalent statement for
equation (6.2) is
We
call f(x) the integrand, and we say
F(x) is equal to the indefinite integral of f(x). The elongated S,
, is the integral sign. This symbol is used because
integration may be shown to be the limit of a sum. INTERPRETATION
OF AN INTEGRAL We
will use the area under a curve for the interpretation of an integral. You
should realize, however, that an integral may represent many things, and it may
be real or abstract. It may represent the plane area, volume, or surface area
of some figure.
Figure 61.Area of a rectangle. 