| 单词 | definite integral |
| 例句 | The definite integral of a vector-valued function is found by finding the definite integrals of the component functions, then putting them back together in a vector-valued function. Calculus, Volume 3 2016-03-30T00:00:00Z In the following exercises, use a change of variables to evaluate the definite integral. Calculus, Vol 2 2016-03-30T00:00:00Z In this section, we look at how to apply the concept of the area under the curve to a broader set of functions through the use of the definite integral. Calculus, Volume 1 2016-03-30T00:00:00Z All such definite integrals as the above are said to be “improper.” Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 3 "Frost" to "Fyzabad" 2011-08-15T02:00:28.473Z But, as in the analogous cases of algebraic equations and definite integrals, it may be quite possible to find, by methods of approximation, an arithmetical solution which is sufficient for the purpose in hand.—Bibliography: The New Gresham Encyclopedia Volume 4, Part 1: Deposition to Eberswalde 2011-04-14T02:00:57.977Z While an integrator determines the value of a definite integral, hence a Integraphs. mere constant, an integraph gives the value of an indefinite integral, which is a function of x. Encyclopaedia Britannica, 11th Edition, Volume 4, Part 4 "Bulgaria" to "Calgary" In the following exercises, compute each definite integral. Calculus, Vol 2 2016-03-30T00:00:00Z Although the notation for indefinite integrals may look similar to the notation for a definite integral, they are not the same. Calculus, Volume 1 2016-03-30T00:00:00Z When the improper definite integral of a function which becomes, or tends to become, infinite, exists, the integral is said to be “convergent.” Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 3 "Frost" to "Fyzabad" 2011-08-15T02:00:28.473Z In this chapter, we study some additional techniques, including some ways of approximating definite integrals when normal techniques do not work. Calculus, Vol 2 2016-03-30T00:00:00Z The instrument now is an integraph giving the value of a definite integral as function of a variable parameter. Encyclopaedia Britannica, 11th Edition, Volume 4, Part 4 "Bulgaria" to "Calgary" Given that we know the Fundamental Theorem of Calculus, why would we want to develop numerical methods for definite integrals? Calculus, Vol 2 2016-03-30T00:00:00Z However, close attention should always be paid to notation so we know whether we’re working with a definite integral or an indefinite integral. Calculus, Volume 1 2016-03-30T00:00:00Z The first inquiry leads directly to the indefinite integral, the second directly to the definite integral. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 3 "Frost" to "Fyzabad" 2011-08-15T02:00:28.473Z For the following exercises, evaluate the definite integrals. Calculus, Vol 2 2016-03-30T00:00:00Z In the rectangular coordinate system, the definite integral provides a way to calculate the area under a curve. Calculus, Vol 2 2016-03-30T00:00:00Z Consequently, rather than evaluate definite integrals of these functions directly, we resort to various techniques of numerical integration to approximate their values. Calculus, Vol 2 2016-03-30T00:00:00Z On a definite integral, above and below the summation symbol are the boundaries of the interval, ⎡a, b⎤. Calculus, Volume 1 2016-03-30T00:00:00Z The representation of a function by means of an infinite product falls clearly under Baire’s method, while the representation by means of a definite integral is analogous to Brod�n’s method. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 3 "Frost" to "Fyzabad" 2011-08-15T02:00:28.473Z Unfortunately, so far, the only tools we have available to calculate the value of a definite integral are geometric area formulas and limits of Riemann sums, and both approaches are extremely cumbersome. Calculus, Volume 1 2016-03-30T00:00:00Z In this section we look at some more powerful and useful techniques for evaluating definite integrals. Calculus, Volume 1 2016-03-30T00:00:00Z Figure 3.14 Trapezoids may be used to approximate the area under a curve, hence approximating the definite integral. Calculus, Vol 2 2016-03-30T00:00:00Z To clarify, we are using the word limit in two different ways in the context of the definite integral. Calculus, Volume 1 2016-03-30T00:00:00Z The key here is to notice that for any particular value of x, the definite integral is a number. Calculus, Volume 1 2016-03-30T00:00:00Z This always happens when evaluating a definite integral. Calculus, Volume 1 2016-03-30T00:00:00Z Figure 5.28 The evaluation of a definite integral can produce a negative value, even though area is always positive. Calculus, Volume 1 2016-03-30T00:00:00Z The trapezoidal rule for estimating definite integrals uses trapezoids rather than rectangles to approximate the area under a curve. Calculus, Vol 2 2016-03-30T00:00:00Z However, if a nonregular partition is used to define the definite integral, it is not sufficient to take the limit as the number of subintervals goes to infinity. Calculus, Volume 1 2016-03-30T00:00:00Z As you become more familiar with integration, you will get a feel for when to use definite integrals and when to use indefinite integrals. Calculus, Volume 1 2016-03-30T00:00:00Z However, until these concepts are cemented in your mind, think carefully about whether you need a definite integral or an indefinite integral and make sure you are using the proper notation based on your choice. Calculus, Volume 1 2016-03-30T00:00:00Z The second is more familiar; it is simply the definite integral. Calculus, Volume 1 2016-03-30T00:00:00Z In addition, a careful examination of Figure 3.15 leads us to make the following observations about using the trapezoidal rules and midpoint rules to estimate the definite integral of a nonnegative function. Calculus, Vol 2 2016-03-30T00:00:00Z Later in this chapter we develop techniques for evaluating definite integrals without taking limits of Riemann sums. Calculus, Volume 1 2016-03-30T00:00:00Z Just as we did before, we can use definite integrals to calculate the net displacement as well as the total distance traveled. Calculus, Volume 1 2016-03-30T00:00:00Z However, using substitution to evaluate a definite integral requires a change to the limits of integration. Calculus, Volume 1 2016-03-30T00:00:00Z Substitution may be only one of the techniques needed to evaluate a definite integral. Calculus, Volume 1 2016-03-30T00:00:00Z On the other hand, the midpoint rule tends to average out these errors somewhat by partially overestimating and partially underestimating the value of the definite integral over these same types of intervals. Calculus, Vol 2 2016-03-30T00:00:00Z However, for now, we can rely on the fact that definite integrals represent the area under the curve, and we can evaluate definite integrals by using geometric formulas to calculate that area. Calculus, Volume 1 2016-03-30T00:00:00Z Figure 5.38 The indicated area can be calculated by evaluating a definite integral using substitution. Calculus, Volume 1 2016-03-30T00:00:00Z The definite integral can be used to calculate net signed area, which is the area above the x-axis less the area below the x-axis. Calculus, Volume 1 2016-03-30T00:00:00Z The component parts of the definite integral are the integrand, the variable of integration, and the limits of integration. Calculus, Volume 1 2016-03-30T00:00:00Z The midpoint rule approximates the definite integral using rectangular regions whereas the trapezoidal rule approximates the definite integral using trapezoidal approximations. Calculus, Vol 2 2016-03-30T00:00:00Z We do this to confirm that definite integrals do, indeed, represent areas, so we can then discuss what to do in the case of a curve of a function dropping below the x-axis. Calculus, Volume 1 2016-03-30T00:00:00Z The properties of definite integrals can be used to evaluate integrals. Calculus, Volume 1 2016-03-30T00:00:00Z The average value of a function can be calculated using definite integrals. Calculus, Volume 1 2016-03-30T00:00:00Z The Fundamental Theorem of Calculus, Part 2 is a formula for evaluating a definite integral in terms of an antiderivative of its integrand. Calculus, Volume 1 2016-03-30T00:00:00Z Figure 1.28 The evaluation of a definite integral can produce a negative value, even though area is always positive. Calculus, Vol 2 2016-03-30T00:00:00Z This is a very important application of the definite integral, and we examine it in more detail later in the chapter. Calculus, Volume 1 2016-03-30T00:00:00Z When using substitution for a definite integral, we also have to change the limits of integration. Calculus, Volume 1 2016-03-30T00:00:00Z Hydrostatic force is only one of the many applications of definite integrals we explore in this chapter. Calculus, Volume 1 2016-03-30T00:00:00Z In Introduction to Integration, we developed the concept of the definite integral to calculate the area below a curve on a given interval. Calculus, Volume 1 2016-03-30T00:00:00Z Figure 1.38 The indicated area can be calculated by evaluating a definite integral using substitution. Calculus, Vol 2 2016-03-30T00:00:00Z These properties, along with the rules of integration that we examine later in this chapter, help us manipulate expressions to evaluate definite integrals. Calculus, Volume 1 2016-03-30T00:00:00Z In the preceding section, we used definite integrals to find the area between two curves. Calculus, Volume 1 2016-03-30T00:00:00Z In this section, we use definite integrals to find volumes of three-dimensional solids. Calculus, Volume 1 2016-03-30T00:00:00Z In this case, we can use a definite integral to calculate the volume of the solid. Calculus, Volume 1 2016-03-30T00:00:00Z We can use numerical integration to estimate the values of definite integrals when a closed form of the integral is difficult to find or when an approximate value only of the definite integral is needed. Calculus, Vol 2 2016-03-30T00:00:00Z Fortunately, we can use a definite integral to find the average value of a function such as this. Calculus, Volume 1 2016-03-30T00:00:00Z In this section, we use definite integrals to find the arc length of a curve. Calculus, Volume 1 2016-03-30T00:00:00Z So, as we have done many times before, we form a partition, a Riemann sum, and, ultimately, a definite integral to calculate the force. Calculus, Volume 1 2016-03-30T00:00:00Z Just as definite integrals can be used to find the area under a curve, they can also be used to find the area between two curves. Calculus, Volume 1 2016-03-30T00:00:00Z Simpson’s rule approximates the definite integral by first approximating the original function using piecewise quadratic functions. Calculus, Vol 2 2016-03-30T00:00:00Z In the following exercises, given Ln or Rn as indicated, express their limits as n → ∞ as definite integrals, identifying the correct intervals. Calculus, Volume 1 2016-03-30T00:00:00Z The method of cylindrical shells is another method for using a definite integral to calculate the volume of a solid of revolution. Calculus, Volume 1 2016-03-30T00:00:00Z The arc length of a curve can be calculated using a definite integral. Calculus, Volume 1 2016-03-30T00:00:00Z Taking a limit then gives us the definite integral formula. Calculus, Volume 1 2016-03-30T00:00:00Z We can use infinite series to evaluate complicated functions, approximate definite integrals, and create new functions. Calculus, Vol 2 2016-03-30T00:00:00Z In the previous two sections, we looked at the definite integral and its relationship to the area under the curve of a function. Calculus, Volume 1 2016-03-30T00:00:00Z Several physical applications of the definite integral are common in engineering and physics. Calculus, Volume 1 2016-03-30T00:00:00Z |
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