Misplaced Pages

#593406

55-466: Astronauts described the gun as easier to use than other methods of maneuvering during space-walking. It provided an impulse to send the space-walker away from and back to the spacecraft, and was the easiest way for them to control their motions in the microgravity environment . The device received its propellant from tanks on the device and used pressurized oxygen to control and propel the astronaut via conservation of momentum . Ed White enjoyed using

110-629: A closed and bounded interval [ a , b ] and can be generalized to other notions of integral (Lebesgue and Daniell). In this section, f is a real-valued Riemann-integrable function . The integral over an interval [ a , b ] is defined if a < b . This means that the upper and lower sums of the function f are evaluated on a partition a = x 0 ≤ x 1 ≤ . . . ≤ x n = b whose values x i are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating f within intervals [ x i  , x i  +1 ] where an interval with

165-456: A , b ] is its width, b − a , so that the Lebesgue integral agrees with the (proper) Riemann integral when both exist. In more complicated cases, the sets being measured can be highly fragmented, with no continuity and no resemblance to intervals. Using the "partitioning the range of f " philosophy, the integral of a non-negative function f  : R → R should be the sum over t of

220-411: A bounded interval, subsequently more general functions were considered—particularly in the context of Fourier analysis —to which Riemann's definition does not apply, and Lebesgue formulated a different definition of integral , founded in measure theory (a subfield of real analysis ). Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed. These approaches based on

275-453: A certain class of "simple" functions, may be used to give an alternative definition of the integral. This is the approach of Daniell for the case of real-valued functions on a set X , generalized by Nicolas Bourbaki to functions with values in a locally compact topological vector space. See Hildebrandt 1953 for an axiomatic characterization of the integral. A number of general inequalities hold for Riemann-integrable functions defined on

330-453: A connection between integration and differentiation . Barrow provided the first proof of the fundamental theorem of calculus . Wallis generalized Cavalieri's method, computing integrals of x to a general power, including negative powers and fractional powers. The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Leibniz and Newton . The theorem demonstrates

385-565: A connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework that both Leibniz and Newton developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions with continuous domains. This framework eventually became modern calculus , whose notation for integrals

440-509: A function f over the interval [ a , b ] is equal to S if: When the chosen tags are the maximum (respectively, minimum) value of the function in each interval, the Riemann sum becomes an upper (respectively, lower) Darboux sum , suggesting the close connection between the Riemann integral and the Darboux integral . It is often of interest, both in theory and applications, to be able to pass to

495-458: A function f with respect to such a tagged partition is defined as thus each term of the sum is the area of a rectangle with height equal to the function value at the chosen point of the given sub-interval, and width the same as the width of sub-interval, Δ i = x i − x i −1 . The mesh of such a tagged partition is the width of the largest sub-interval formed by the partition, max i =1... n Δ i . The Riemann integral of

550-402: A higher index lies to the right of one with a lower index. The values a and b , the end-points of the interval , are called the limits of integration of f . Integrals can also be defined if a > b : With a = b , this implies: The first convention is necessary in consideration of taking integrals over subintervals of [ a , b ] ; the second says that an integral taken over

605-414: A letter to Paul Montel : I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay

SECTION 10

#1732855365594

660-452: A suitable class of functions (the measurable functions ) this defines the Lebesgue integral. A general measurable function f is Lebesgue-integrable if the sum of the absolute values of the areas of the regions between the graph of f and the x -axis is finite: In that case, the integral is, as in the Riemannian case, the difference between the area above the x -axis and the area below

715-432: A tank to be carried on the astronaut's back. Astronaut David Scott never got a chance to use it, because the mission had to be terminated before his EVA due to a critical thruster problem. The Gemini 10 device used by Michael Collins received its nitrogen gas propellant from inside the spacecraft, through a hose bundled with the astronaut's umbilical connector. Collins successfully used it to move back and forth between

770-471: A varying force is the integral of the force F with respect to time: J = ∫ F d t . {\displaystyle \mathbf {J} =\int \mathbf {F} \,\mathrm {d} t.} The SI unit of impulse is the newton second (N⋅s), or the Cupp, and the dimensionally equivalent unit of momentum is the kilogram metre per second (kg⋅m/s). The corresponding English engineering unit

825-411: A wide variety of scientific fields thereafter. A definite integral computes 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

880-457: Is a step change , and is not physically possible. However, this is a useful model for computing the effects of ideal collisions (such as in videogame physics engines ). Additionally, in rocketry, the term "total impulse" is commonly used and is considered synonymous with the term "impulse". The application of Newton's second law for variable mass allows impulse and momentum to be used as analysis tools for jet - or rocket -propelled vehicles. In

935-418: Is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval. A tagged partition of a closed interval [ a , b ] on the real line is a finite sequence This partitions the interval [ a , b ] into n sub-intervals [ x i −1 , x i ] indexed by i , each of which is "tagged" with a specific point t i ∈ [ x i −1 , x i ] . A Riemann sum of

990-507: Is drawn directly from the work of Leibniz. While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of rigour . Bishop Berkeley memorably attacked the vanishing increments used by Newton, calling them " ghosts of departed quantities ". Calculus acquired a firmer footing with the development of limits . Integration was first rigorously formalized, using limits, by Riemann . Although all bounded piecewise continuous functions are Riemann-integrable on

1045-501: Is not uncommon to leave out dx when only the simple Riemann integral is being used, or the exact type of integral is immaterial. For instance, one might write ∫ a b ( c 1 f + c 2 g ) = c 1 ∫ a b f + c 2 ∫ a b g {\textstyle \int _{a}^{b}(c_{1}f+c_{2}g)=c_{1}\int _{a}^{b}f+c_{2}\int _{a}^{b}g} to express

1100-449: Is of great importance to have a definition of the integral that allows a wider class of functions to be integrated. Such an integral is the Lebesgue integral, that exploits the following fact to enlarge the class of integrable functions: if the values of a function are rearranged over the domain, the integral of a function should remain the same. Thus Henri Lebesgue introduced the integral bearing his name, explaining this integral thus in

1155-492: Is the pound -second (lbf⋅s), and in the British Gravitational System , the unit is the slug -foot per second (slug⋅ft/s). Impulse J produced from time t 1 to t 2 is defined to be J = ∫ t 1 t 2 F d t , {\displaystyle \mathbf {J} =\int _{t_{1}}^{t_{2}}\mathbf {F} \,\mathrm {d} t,} where F

SECTION 20

#1732855365594

1210-746: Is the change in linear momentum from time t 1 to t 2 . This is often called the impulse-momentum theorem (analogous to the work-energy theorem ). As a result, an impulse may also be regarded as the change in momentum of an object to which a resultant force is applied. The impulse may be expressed in a simpler form when the mass is constant: J = ∫ t 1 t 2 F d t = Δ p = m v 2 − m v 1 , {\displaystyle \mathbf {J} =\int _{t_{1}}^{t_{2}}\mathbf {F} \,\mathrm {d} t=\Delta \mathbf {p} =m\mathbf {v_{2}} -m\mathbf {v_{1}} ,} where Impulse has

1265-448: Is the continuous analog of a sum , which is used to calculate areas , volumes , and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus , the other being differentiation . Integration was initially used to solve problems in mathematics and physics , such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to

1320-420: Is the given function; in this case, they are also called indefinite integrals . The fundamental theorem of calculus relates definite integration to differentiation and provides a method to compute the definite integral of a function when its antiderivative is known; differentiation and integration are inverse operations. Although methods of calculating areas and volumes dated from ancient Greek mathematics ,

1375-1013: Is the resultant force applied from t 1 to t 2 . From Newton's second law , force is related to momentum p by F = d p d t . {\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}.} Therefore, J = ∫ t 1 t 2 d p d t d t = ∫ p 1 p 2 d p = p 2 − p 1 = Δ p , {\displaystyle {\begin{aligned}\mathbf {J} &=\int _{t_{1}}^{t_{2}}{\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}\,\mathrm {d} t\\&=\int _{\mathbf {p} _{1}}^{\mathbf {p} _{2}}\mathrm {d} \mathbf {p} \\&=\mathbf {p} _{2}-\mathbf {p} _{1}=\Delta \mathbf {p} ,\end{aligned}}} where Δ p

1430-422: The ancient Greek astronomer Eudoxus and philosopher Democritus ( ca. 370 BC), which sought to find areas and volumes by breaking them up into an infinite number of divisions for which the area or volume was known. This method was further developed and employed by Archimedes in the 3rd century BC and used to calculate the area of a circle , the surface area and volume of a sphere , area of an ellipse ,

1485-430: The differential of the variable x , indicates that the variable of integration is x . The function f ( x ) is called the integrand, the points a and b are called the limits (or bounds) of integration, and the integral is said to be over the interval [ a , b ] , called the interval of integration. A function is said to be integrable if its integral over its domain is finite. If limits are specified,

1540-430: The x -axis: where Although the Riemann and Lebesgue integrals are the most widely used definitions of the integral, a number of others exist, including: The collection of Riemann-integrable functions on a closed interval [ a , b ] forms a vector space under the operations of pointwise addition and multiplication by a scalar, and the operation of integration is a linear functional on this vector space. Thus,

1595-513: The Gemini and the Agena Target Vehicle . Richard Gordon did not get to use his HHMU on Gemini 11 , because his EVA had to be cut short when he became fatigued. Impulse (physics) In classical mechanics , impulse (symbolized by J or Imp ) is the change in momentum of an object. If the initial momentum of an object is p 1 , and a subsequent momentum is p 2 ,

1650-484: The area under a parabola , the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral . A similar method was independently developed in China around the 3rd century AD by Liu Hui , who used it to find the area of the circle. This method was later used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find

1705-442: The areas between a thin horizontal strip between y = t and y = t + dt . This area is just μ { x  : f ( x ) > t }  dt . Let f ( t ) = μ { x  : f ( x ) > t } . The Lebesgue integral of f is then defined by where the integral on the right is an ordinary improper Riemann integral ( f is a strictly decreasing positive function, and therefore has a well-defined improper Riemann integral). For

Hand-held maneuvering unit - Misplaced Pages Continue

1760-477: The box notation was difficult for printers to reproduce, so these notations were not widely adopted. The term was first printed in Latin by Jacob Bernoulli in 1690: "Ergo et horum Integralia aequantur". In general, the integral of a real-valued function f ( x ) with respect to a real variable x on an interval [ a , b ] is written as The integral sign ∫ represents integration. The symbol dx , called

1815-482: The case of rockets, the impulse imparted can be normalized by unit of propellant expended, to create a performance parameter, specific impulse . This fact can be used to derive the Tsiolkovsky rocket equation , which relates the vehicle's propulsive change in velocity to the engine's specific impulse (or nozzle exhaust velocity) and the vehicle's propellant- mass ratio . Integral In mathematics , an integral

1870-470: The collection of integrable functions is closed under taking linear combinations , and the integral of a linear combination is the linear combination of the integrals: Similarly, the set of real -valued Lebesgue-integrable functions on a given measure space E with measure μ is closed under taking linear combinations and hence form a vector space, and the Lebesgue integral is a linear functional on this vector space, so that: More generally, consider

1925-534: The definite integral, with limits above and below the integral sign, was first used by Joseph Fourier in Mémoires of the French Academy around 1819–1820, reprinted in his book of 1822. Isaac Newton used a small vertical bar above a variable to indicate integration, or placed the variable inside a box. The vertical bar was easily confused with . x or x ′ , which are used to indicate differentiation, and

1980-456: The foundations of modern calculus, with Cavalieri computing the integrals of x up to degree n = 9 in Cavalieri's quadrature formula . The case n = −1 required the invention of a function , the hyperbolic logarithm , achieved by quadrature of the hyperbola in 1647. Further steps were made in the early 17th century by Barrow and Torricelli , who provided the first hints of

2035-459: The gun and found it useful, but quickly ran out of propellant, forcing him to pull on his tether to continue maneuvers. However, fellow crewman James McDivitt recalled the gun as being "hopeless" and "utterly useless" as it required precise aim through the user's center of mass in order to translate in a straight line without inducing unwanted rotation . The device carried on Gemini 8 (March 16–17, 1966) received its Freon 14 propellant from

2090-534: The integral is called a definite integral. When the limits are omitted, as in the integral is called an indefinite integral, which represents a class of functions (the antiderivative ) whose derivative is the integrand. The fundamental theorem of calculus relates the evaluation of definite integrals to indefinite integrals. There are several extensions of the notation for integrals to encompass integration on unbounded domains and/or in multiple dimensions (see later sections of this article). In advanced settings, it

2145-413: The integration is performed. For example, a line integral is defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting two points in space. In a surface integral , the curve is replaced by a piece of a surface in three-dimensional space . The first documented systematic technique capable of determining integrals is the method of exhaustion of

2200-429: The limit under the integral. For instance, a sequence of functions can frequently be constructed that approximate, in a suitable sense, the solution to a problem. Then the integral of the solution function should be the limit of the integrals of the approximations. However, many functions that can be obtained as limits are not Riemann-integrable, and so such limit theorems do not hold with the Riemann integral. Therefore, it

2255-434: The linearity holds for the subspace of functions whose integral is an element of V (i.e. "finite"). The most important special cases arise when K is R , C , or a finite extension of the field Q p of p-adic numbers , and V is a finite-dimensional vector space over K , and when K = C and V is a complex Hilbert space . Linearity, together with some natural continuity properties and normalization for

Hand-held maneuvering unit - Misplaced Pages Continue

2310-518: The linearity of the integral, a property shared by the Riemann integral and all generalizations thereof. Integrals appear in many practical situations. For instance, from the length, width and depth of a swimming pool which is rectangular with a flat bottom, one can determine the volume of water it can contain, the area of its surface, and the length of its edge. But if it is oval with a rounded bottom, integrals are required to find exact and rigorous values for these quantities. In each case, one may divide

2365-685: The number of pieces increases to infinity, it will reach a limit which is the exact value of the area sought (in this case, 2/3 ). One writes which means 2/3 is the result of a weighted sum of function values, √ x , multiplied by the infinitesimal step widths, denoted by dx , on the interval [0, 1] . There are many ways of formally defining an integral, not all of which are equivalent. The differences exist mostly to deal with differing special cases which may not be integrable under other definitions, but are also occasionally for pedagogical reasons. The most commonly used definitions are Riemann integrals and Lebesgue integrals. The Riemann integral

2420-415: The object has received an impulse J : J = p 2 − p 1 . {\displaystyle \mathbf {J} =\mathbf {p} _{2}-\mathbf {p} _{1}.} Momentum is a vector quantity, so impulse is also a vector quantity. Newton’s second law of motion states that the rate of change of momentum of an object is equal to the resultant force F acting on

2475-455: The object: F = p 2 − p 1 Δ t , {\displaystyle \mathbf {F} ={\frac {\mathbf {p} _{2}-\mathbf {p} _{1}}{\Delta t}},} so the impulse J delivered by a steady force F acting for time Δ t is: J = F Δ t . {\displaystyle \mathbf {J} =\mathbf {F} \Delta t.} The impulse delivered by

2530-402: 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

2585-477: The real number system are the ones most common today, but alternative approaches exist, such as a definition of integral as the standard part of an infinite Riemann sum, based on the hyperreal number system. The notation for the indefinite integral was introduced by Gottfried Wilhelm Leibniz in 1675. He adapted the integral symbol , ∫ , from the letter ſ ( long s ), standing for summa (written as ſumma ; Latin for "sum" or "total"). The modern notation for

2640-455: 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 general than Riemann's in the sense that a wider class of functions are Lebesgue-integrable. Integrals may be generalized depending on the type of the function as well as the domain over which

2695-417: The results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid . The next significant advances in integral calculus did not begin to appear until the 17th century. At this time, the work of Cavalieri with his method of indivisibles , and work by Fermat , began to lay

2750-409: The right end height of each piece (thus √ 0 , √ 1/5 , √ 2/5 , ..., √ 1 ) and sum their areas to get the approximation which is larger than the exact value. Alternatively, when replacing these subintervals by ones with the left end height of each piece, the approximation one gets is too low: with twelve such subintervals the approximated area is only 0.6203. However, when

2805-487: The same units and dimensions (MLT ) as momentum. In the International System of Units , these are kg ⋅ m/s = N ⋅ s . In English engineering units , they are slug ⋅ ft/s = lbf ⋅ s . The term "impulse" is also used to refer to a fast-acting force or impact . This type of impulse is often idealized so that the change in momentum produced by the force happens with no change in time. This sort of change

SECTION 50

#1732855365594

2860-441: The several heaps one after the other to the creditor. This is my integral. As Folland puts it, "To compute the Riemann integral of f , one partitions the domain [ a , b ] into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of f ". The definition of the Lebesgue integral thus begins with a measure , μ. In the simplest case, the Lebesgue measure μ ( A ) of an interval A = [

2915-421: The sought quantity into infinitely many infinitesimal pieces, then sum the pieces to achieve an accurate approximation. As another example, to find the area of the region bounded by the graph of the function f ( x ) = x {\textstyle {\sqrt {x}}} between x = 0 and x = 1 , one can divide the interval into five pieces ( 0, 1/5, 2/5, ..., 1 ), then construct rectangles using

2970-414: The vector space of all measurable functions on a measure space ( E , μ ) , taking values in a locally compact complete topological vector space V over a locally compact topological field K , f  : E → V . Then one may define an abstract integration map assigning to each function f an element of V or the symbol ∞ , that is compatible with linear combinations. In this situation,

3025-671: The volume of a sphere. In the Middle East, Hasan Ibn al-Haytham, Latinized as Alhazen ( c.  965  – c.  1040  AD) derived a formula for the sum of fourth powers . Alhazen determined the equations to calculate the area enclosed by the curve represented by y = x k {\displaystyle y=x^{k}} (which translates to the integral ∫ x k d x {\displaystyle \int x^{k}\,dx} in contemporary notation), for any given non-negative integer value of k {\displaystyle k} . He used

#593406