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Calculus : IntegrateThe integrate command will calculate definite and indefinite integrals. In order to use the basic integrate command to find an indefinite integral, simply enter the expression you wish to integrate, enter the variable to integrate with respect to and click the Integrate button. If you wish to find the definite integral between two limits, a and b say, enter x,a,b in the variables text area. The advanced integrate command allows you to calculate multiple integrals. Simply enter each variable you wish to integrate with respect to (with limits as described above if required) on a separate line in the variables text area. ExamplesHere are some examples illustrating the types of expressions you can use the integrate command on and the results which QuickMath will return. Basic integrate command
Advanced integrate command
The Integral as an AreaThe definite integral, introduced in this section, and the derivative are probably the two fundamental concepts in calculus. We shall introduce the definite integral in terms of the geometrical notion of "area under a curve." The familiar geometrical concept of area provides an intuitive way to approach the integral. However, the definite integral has many other applications. In the next chapter, we shall see applications to business, science, and probability and statistics. The tiein between the antiderivative and the integral will become more apparent in the next section.
We are about to introduce what may seem to be rather curious notation. (The objects we are now defining will ultimately have more application than area, and we need a sufficiently general notation.) Let/be a function satisfying Conditions 1 and 2. For any x in [a, b ], the area under the graph off from a to x (see Figure 2) is denoted by
Warning! Although the notations for both the integral and the antiderivative include the integration symbol, they should not be confused; the underlying concepts the represent are quite distinct. We shall see later an amazing relationship between these concepts. Note that the integral depends on x (as well as a and f, of course); thus, to keep the bookkeeping straight, the function f will be given in terms of some variable other than x such as t , for example, in the above definition and it will be convenient to think of the horizontal axis as, in this case, a taxis (see Figure 2). In terms of this new notation: Example 1 Interpret as an area and compute its value.
integration of f adds areas of regions that lie above the xaxis and subtracts areas of regions that lie below the xaxis. For example, suppose f is the function graphed in Figure 4. The area under the graph of f from a to c is equal to the area under the graph from c to b; suppose each equals A square units. Since the graph of f lies below the xaxis from c to b , this area is subtracted when we evaluate the integral, and hence
