Definite Integral
A Definite Integral is an integral that represents the area under a curve over a specific interval on the x-axis.
- Context:
- It can be used to calculate the total accumulated change of a quantity, such as distance or work done, over a specified interval.
- It can be referenced in fields, such as physics, engineering, and economics.
- …
- Example(s):
- Finding the area between two curves, as shown in the earlier response.
- Calculating the work done by a force when displacement occurs along a curve.
- …
- Counter-Example(s):
- Indefinite Integral: which does not have specified limits of integration.
- See: Fundamental Theorem of Calculus, Integral, Antiderivative, Integral Approximation.
References
2023
- A definite integral is a mathematical concept in calculus that represents the area under a curve over a specific interval on the x-axis. It is a special type of integral that has both lower and upper limits of integration, which are the starting and ending points of the interval, respectively.
- The definite integral of a function f(x) from a to b is denoted as: ∫[a, b] f(x) dx
- The process of finding the definite integral involves finding the antiderivative (also known as the indefinite integral) of the function f(x) and then applying the Fundamental Theorem of Calculus. The theorem states that if F(x) is the antiderivative of f(x), then the definite integral of f(x) from a to b is equal to F(b) - F(a).
- In the case of finding the area between two curves, the definite integral is used to compute the difference between the two functions over a specified interval, which represents the area between the curves.
2023
- (Wikipedia, 2023) ⇒ https://en.wikipedia.org/wiki/integral Retrieved:2023-4-26.
- … The integrals enumerated here are those termed definite integrals, which can be interpreted as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function; in this case, they are also called indefinite integrals. The fundamental theorem of calculus relates definite integrals with differentiation and provides a method to compute the definite integral of a function when its antiderivative is known.
Although methods of calculating areas and volumes dated from ancient Greek mathematics, the principles of integration were formulated independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of infinitesimal width. Bernhard Riemann later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into infinitesimally thin vertical slabs. In the early 20th century, Henri Lebesgue generalized Riemann's formulation by introducing what is now referred to as the Lebesgue integral; it is more robust than Riemann's in the sense that a wider class of functions are Lebesgue-integrable. ...
- … The integrals enumerated here are those termed definite integrals, which can be interpreted as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function; in this case, they are also called indefinite integrals. The fundamental theorem of calculus relates definite integrals with differentiation and provides a method to compute the definite integral of a function when its antiderivative is known.
2023
- (Wikipedia, 2023) ⇒ https://en.wikipedia.org/wiki/List_of_definite_integrals Retrieved:2023-4-26.
- In mathematics, the definite integral : [math]\displaystyle{ \int_a^b f(x)\, dx }[/math] is the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the lines x = a and x = b, such that area above the x-axis adds to the total, and that below the x-axis subtracts from the total.
The fundamental theorem of calculus establishes the relationship between indefinite and definite integrals and introduces a technique for evaluating definite integrals.
If the interval is infinite the definite integral is called an improper integral and defined by using appropriate limiting procedures. for example: : [math]\displaystyle{ \int_a^\infty f(x)\, dx=\lim_{b\to\infty}\left[\int_a^b f(x)\, dx\right] }[/math] A constant, such pi, that may be defined by the integral of an algebraic function over an algebraic domain is known as a period.
The following is a list of some of the most common or interesting definite integrals. For a list of indefinite integrals see List of indefinite integrals.
- In mathematics, the definite integral : [math]\displaystyle{ \int_a^b f(x)\, dx }[/math] is the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the lines x = a and x = b, such that area above the x-axis adds to the total, and that below the x-axis subtracts from the total.