Integral. For example, consider the curve y = f(x) between x = 0 and x = 1, with f(x) = x. We ask:

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1 Integral Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the integral is defined informally to be the net signed area of the region in the xy-plane bounded by the graph of ƒ, the x-axis, and the vertical lines x = a and x = b. The term "integral" may also refer to the notion of antiderivative, a function F whose derivative is the given function ƒ. In this case it is called an indefinite integral, while the integrals discussed in this article are termed definite integrals. Some authors maintain a distinction between antiderivatives and indefinite integrals. The principles of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late seventeenth century. Through the fundamental theorem of calculus, which they independently developed, integration is connected with differentiation: if ƒ is a continuous real-valued function defined on a closed interval [a, b], then, once an antiderivative F of ƒ is known, the definite integral of ƒ over that interval is given by For example, consider the curve y = f(x) between x = 0 and x = 1, with f(x) = x. We ask: What is the area under the function f, in the interval from 0 to 1? and call this (yet unknown) area the integral of f. The notation for this integral will be As a first approximation, look at the unit square given by the sides x = 0 to x = 1 and y = f(0) = 0 and y = f(1) = 1. Its area is exactly 1. As it is, the true value of the integral must be somewhat less. Decreasing the width of the approximation rectangles shall give a better result; so cross the interval in five steps, using the approximation points 0, 1 5, 2 5, and so on to 1. Fit a box for each step using the right end height of each curve piece, thus 1 5, 2 5, and so on to 1 = 1. Summing the areas of these rectangles, we get a better approximation for the sought integral, namely Notice that we are taking a sum of finitely many function values of f, multiplied with the differences of two subsequent approximation points. We can easily see that the approximation is still too large. Using more steps produces a closer approximation, but will never be exact: replacing the 5 subintervals by twelve as depicted, we will get an

2 approximate value for the area of , which is too small. The key idea is the transition from adding finitely many differences of approximation points multiplied by their respective function values to using infinitely fine, or infinitesimal steps. As for the actual calculation of integrals, the fundamental theorem of calculus, due to Newton and Leibniz, is the fundamental link between the operations of differentiating and integrating. Applied to the square root curve, f(x) = x 1/2, it says to look at the antiderivative F(x) = 2 3 x 3/2, and simply take F(1) F(0), where 0 and 1 are the boundaries of the interval [0,1]. (This is a case of a general rule, that for f(x) = x q, with q 1, the related function, the so-called antiderivative is F(x) = (x q+1 )/(q + 1).) So the exact value of the area under the curve is computed formally as There are many useful integral equations, belows are basic integral equation for ECON206 class. 1) x dx x 2) kx dx k x dx k 3) if b a, c, then fxdx b a x fxdx k b a fxdx Exponential Function The exponential function e x can be defined, in a variety of equivalent ways, as an infinite series. In particular it may be defined by a power series: It is also the following limit:.

3 And e 1 = e = The basic property of exponential function is And the most important property of exponential function is This implies that e dx e Logarithm The logarithm of x to the base b is written log b (x) or, if the base is implicit, as log(x). So, for a number x, a base b and an exponent y, An important feature of logarithms is that they reduce multiplication to addition, by the formula: If b=e(exponential), let log a ln a and we call it natural log. Since exponential and logarithmic functions of the same base are inverses of one another, if you compose the two functions together, they will cancel one another out.

4 Since you will see common and natural logs most often, here is that inverse relationship expressed in terms of their respective bases: Formally, ln(a) may be defined as the area under the graph of 1/x from 1 to a, that is as the integral, This defines a logarithm because it satisfies the fundamental property of a logarithm: This can be demonstrated by letting as follows: The number e can then be defined as the unique real number a such that ln(a) = 1. Since logarithmic and exponential functions are one another's inverses, it is easy to construct the graph of any logarithmic function y = log a x based on the corresponding graph of y = a x. Graphs of inverse functions are reflections of one another across the line y = x, since each graph contains the coordinates of the other graph, with each coordinate pair reversed. It is no surprise, then, that because all exponential graphs of the form y = a x contain the point (0,1), then all logarithmic graphs of the form y = log a x contain the point (1,0). In Figure 1, you can visually verify that the graphs of the natural logarithmic and natural exponential functions are, indeed, reflections of one another about the line y = x.

5 Figure 1 The graphs of y = e x and y = in x are reflections of one another about the line y = x, as are all inverse functions. Note that the domain of ln x, like all logarithmic functions of form y = log a x, is (0, ). Although it might appear that the y values of the logarithmic graph level out, as if approaching a horizontal asymptote, they do not. In fact, a logarithmic graph will grow infinitely tall, albeit much, much slower than its sister the exponential function. A range of (, ) for the logarithmic functions makes sense, since their inverses are exponential functions and have domains of (, ). As we can find that ln2<1 and ln4>1 by Figure 1 and recall that e 1 =a ln a=1 so, 2<e<4 (actually e= )