antiderivative和indefinite integral的区别

　　integral　　英 [u02c8u026antu026agru0259l] 美 [u02c8u026antu026aɡru0259l, u026anu02c8tu025bɡru0259l]　　adj.积分的; 完整的; 必须的;　　n.积分; 整体;　　[例句]So, that"s the general pattern.When we have a double integral to compute, first we want to set it up carefully.　　这就是计算二重积分的一般过程，计算一个二重积分，首先要细心的把它写出来。　　[其他]复数：integrals 形近词： integron integrin

积分

an indispensable part

Integral calculus

Integral在数学上的中文译名为积分，即累积的分数。在天文学释义是新一代γ射线望远镜的名字。数学释义：积分，即累积的分数。网上，有很多的积分活动，如电子邮箱积分，qq秀积分等：生活中，有会员卡积分等等优惠活动。在微积分中积分是微分的逆运算，即知道了函数的导函数，反求原函数。在应用上，积分作用不仅如此，它被大量应用于求和，通俗的说是求曲边三角形的面积，这巧妙的求解方法是积分特殊的性质决定的。一个函数的不定积分（亦称原函数）指另一族函数，这一族函数的导函数恰为前一函数。其中：[F(x) + C]" = f(x)一个实变函数在区间[a,b]上的定积分，是一个实数。它等于该函数的一个原函数在b的值减去在a的值。integral 用法和例句：Such machines cannot truly make computing an integral , invisible part of the way people live their lives .这些机器不能够真正的将计算变成人们生活方式中的一个整体的，无形的部分。This fourth level is the witness of all the other states , the integral awareness that links the other three levels together .这第四层见证其他三种状态，是将其他三个层次连接起来的整体意识。Adding up a whole bunch of small pieces is called " an integral " .把一大堆小的一片一片合在一起叫“积分”。

根据词根记忆法来记忆，意思是完整的，不可或缺的。

一个主要的部分的英文：an integral part of参考例句：An integral part of...的一个主要的部分integral是什么意思：adj. 固有的，基本的，完整的，组成的，集成的n. 整体，整数，积分vertical integration.包括从生产到销售各个阶段的纵向联合。an integral equation 积分方程integrate a restaurant取消饭馆用餐的种族隔离 integral bumper整体式保险杠 The Integral domain and integral pattern in EMT electromagnetic field.介绍了EMT电磁场中的积分域和积分形式问题。 part是什么意思：n. 部分；零件；角色v. （使）分裂；分岔；（使）分开adv. 部分地adj. 部分的sustain the part of恰当地扮演 An aliquot part.整除数，约数能整除的部分。The parts interlock.各部分相互扣住。They are part of evolution and evolution is a part of nature. 它们是发展的一部分，而发展又是自然的一部分。It is made part of iron and part of wood. 它部分由铁部分由木而制成。

没什么区别，请看以下牛津字典对integral这个义项的解释: be an essential part of sth.请参考！

complete彻底的，完整的，已完成的，完成，使完整integral构成整体所必需的，完整的，数学上的积分，完整，部分

你好！日常使用中，essential主要来表达必要的意思，integral这个词更多是做为数学名词：积分，或者是整体的意思如有疑问，请追问。

int型

出现"A and B must be floating-point scalars"错误通常是因为integral函数中的积分上下限输入有误。请确保积分上下限的输入为浮点数，例如：cssCopy code% 定义被积函数f = @(x) x.^2;% 定义积分下限a和上限ba = 0.5;b = 2.5;% 使用integral计算积分result = integral(f, a, b);请注意，integral函数的第一个参数应该是被积函数的函数句柄，而不是被积函数本身。另外，积分上下限的输入顺序应该是先下限，后上限。如果问题仍然存在，请检查代码的其他部分是否有任何错误或潜在的问题。在Matlab中，使用integral函数求解积分时，A和B必须是浮点标量。根据您提供的代码，可能是因为您没有正确定义A和B而导致这个错误。请检查您的代码，确保在调用integral函数之前已正确定义A和B，并且它们都是浮点标量。另外，还有一些其他的问题需要注意：在计算EI和kGA时，需要使用Matlab中的乘法运算符“*”，而不是“（）”。在定义dq(x)时，应该使用函数句柄“q”而不是字符串“"q(x)"”。下面是更新后的代码，您可以参考一下：clearsyms c11 c12 c13 c14syms x nL=280;bb=6;hb=6;Eb=34500000;vb=0.2;EI=(1/6)Eb(pi*(bb^4)/64-pi*(5.4^4)/64);kGA=(Eb/2/(1+vb))(pi(bb^2)/4-pi*(5.4^2)/4)/21;ks=7000;l1=20;l2=260;N=10;fun=@(x)0.05631888exp(-(((x-140)/14.8-(5.90329E-06))/0.019791938)^2)ks;A0=integral(fun,x,l1,l2)/L;An=(2/L)integral(funcos(npix/L),x,l1,l2);q(x)=A0+ symsum((An)cos(npi*x/L),n,1,N);dq(x)=diff(q(x));

integral2 函数没有 "ArrayValued" 选项，应该是不能向量化积分的。不知道你解决了吗，最近也遇到这种计算，用循环的话，感觉计算效率有点不能接受。

table of integral/Lists of integrals

积分符号∫，来自于summa这个希腊单词的首字母，可以简化读作sum。 当然，事实上很多人都读作“积分”，“……的积分”或“对……积分”，包括目前所有教过我的老师。 外国读作“integral”，“the integral of”等。 所以，其实这个东西就是一个约定俗成的读法，怎么读都行。（可参考数学老师对各种希腊字母的各种方言读法:-D）

问题一：银行卡的“积分”，这里的“积分”，用英语该怎么说？ 给你问了一个在外企银行工作的朋友，她说bonus points或者reward points都可以，简写一般是BP。 问题二：“积分兑换”用英语怎么说 “积分兑换”用英语 credits exchange 问题三：会计中的积分制用英语怎么说 会计中的积分制 Integral system in accounting 会计中的积分制 Integral system in accounting 问题四：求导和积分用英语怎么说 Derivation and integration 例句： 仅用幂函数的求导和积分的形式运算公式和初等的方法求出了自然数有限项幂和当幂为非负整数时的求和公式。 This paper gives the formulas of the sums when the power is non-negative integersthrough the derivation and integration of the power function and by using theelementary method. 问题五：积分兑换商品的 “积分” 用英文怎么讲？ credit 问题六：信用卡积分英文怎么说？ point（s） of credit card信用卡积分 比如：If you join our membership, you can earn 1 point储member"s credit for each dollar you spend. 如果您成为会员，您每消费一美元,就会获得一分积分。 问题七：积分兑换英语怎么说？ Integral Exchange.

标示单位向量n of S作为从媒介1到媒介2的常量，对于该封闭曲面用麦克斯韦方程进行积分“从媒介一到媒介二”“a small cylinder”无法落实更具体的翻译n of S 粗糙的理解也许是在S平面内的n向量（不靠谱）单词注解：denote vt.预示，意味，标示vector n.矢量，航线unit vector 单位向量medium n.媒介integral n.积分，整体surface integral 面积分maxwell"equations 麦克斯韦方程（关于这个http://zhidao.baidu.com/question/212905555.html）closed surface 封闭曲面cylinder 圆柱，圆筒，圆柱状物

integral 是总放热（吸热）量；normalized 是标准放热（吸热）量 J/g；所以你的问题是：总吸热量为314mj；单位吸热量为 47J/g；

calculous

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make occasional use of teaching aids such as computers and tape recorders，从teaching aids可知是辅助工作

printPreviewDialog1.PrintPreviewControl.StartPage = 0; class Composite : Component { private List children = new List(); public Composite(string name) : base(name) { } public override void Add(Component c) { children.Add(c); } public override void Remove(Component c) { children.Remove(c); }

function mainclcclearr=1;Neph=0.0005*integral(@(rzc) 5.3184*gax(rzc).*grad(rzc,r).*exp(-0.0073*rzc).*exp(1.7949*(rzc-0.01213)),1,10, "ArrayValued",true);aa=double(Neph);qiao=1;endfunction f=gax(rzc)f=1./(pi*exp(-0.0073*(rzc-0.01213))).*integral(@(qzc) exp(-0.0073*(rzc-0.01213)./cos(qzc)),0,pi/2);endfunction f=grad(rzc,r)f=1./(pi*exp(-0.0073*(rzc-0.01213))).*integral(@(qzc) exp(-0.0073*sqrt(rzc.^2+0.01213^2-2*rzc*r*cos(qzc))),0,acos(0.01213/rzc));end这是用子函数实现的方法，嵌套函数对于复杂函数很不方便，必须注意是你式子里面变量r没有定义，我随便设置r=1，你可以根据你的计算设置对应数值觉得有用就给个赞哦

在 MATLAB 中，可以使用 integral 函数来对给定的输入输出曲线进行积分。integral 函数的基本语法如下：integral(fun, a, b)其中，fun 是一个函数句柄，表示要进行积分的函数；a 和 b 分别表示积分的下限和上限。例如，如果有一个输入输出曲线的函数表达式为 y = f(x)，要对其进行积分，可以按以下步骤进行：定义输入输出曲线的函数表达式 f(x)，可以通过函数句柄的方式定义一个匿名函数，例如：f = @(x) x.^2 + 3*x + 5; % 这里假设输入输出曲线为 y = x^2 + 3x + 5这里使用了 .^ 表示对矩阵元素进行逐元素乘方运算，避免了矩阵乘方运算的错误。使用 integral 函数对函数 f(x) 进行积分，指定积分的下限和上限，例如：a = 0; % 积分下限b = 1; % 积分上限result = integral(f, a, b);这里 result 将保存计算得到的积分值。

integral multipleun.整数倍整倍数；整数倍数例句1.Each tender bid must be for an integral multiple of the minimum denomination of the Notes.每项投标必须为债券最低面值的整数倍数。2.If you think a file contains an integral multiple of four bytes, then check that.如果认为文件包含一个四字节的整数倍的数据，请验证一下。3.Radiation of the Integral Multiple Semi-wave Antenna整数倍半波天线的辐射4.Its frequency is an integral multiple of one-half the classical frequency of revolution of the electron in the bound orbit.其频率为在束缚轨道上电子旋转的经典频率之半的整数倍。5.If you want to specify a page size, the size must be an integral multiple of the default page size, not greater than 16 kilobytes.如果希望指定页大小，指定的值必须是默认页大小的整数倍，而且不能超过16KB。

report中的这些都是输出方式：intergral是积分 一般常用area-weighted 来输出表示的是面平均值 能量守恒要在fluent-report-energy项 全部选中后就能够看到最后能量的误差值 这个值越小越好

double integral[英][u02c8du028cbl u02c8intiɡru0259l][美][u02c8du028cbu0259l u02c8u026antu026aɡru0259l]二重插值，二重积分，重积分; 双语例句1Now, how do we compute that double integral?那么，怎么计算这个二重积分呢？2I will just get double integral over R of x dA, which looks certainly a lot more pleasant.我只需要计算xdA在R上的二重积分，看上去确实简单许多。

integral (x^2+1)/(x^2-2x+2)dx=integrate[dx +d((X-1)^2+1)/((X-1)^2+1)+1/((X-1)^2+1)dx]=x+ln((X-1)^2+1)+Arctan(X-1)+c

奉贤精锐英语老师为你解答：积分学

integer是整型的意思，但是C++里没有这个数据类型，只有int型，在java里有integer，Integer 是一个类，是对象类型 int是原始类型。 简单说来就是 int 的别名，再深层次个人觉得没太大必要知道 integer英文意思“整数”pascal,java,VB等等编程软件都把integer作数据类型c++里的整型数据类型不是integer，是其缩写int。实际编程时基本不会出现integer

matlab求解椭圆积分，可以用数值积分函数（如integral， trapz）来求解。解决的办法：1、首先建立自定义的椭圆积分函数fun1=@(x)1./sqrt(1-q^2*sin(x).^2); %F(a,q)fun2=@(x)sqrt(1-q^2*sin(x).^2); %E(a,q)2、确定a，q值3、使用integral函数求解。4、完整的代码a=0.5;q=0.2;fun1=@(x)1./sqrt(1-q^2*sin(x).^2); %F(a,q)fun2=@(x)sqrt(1-q^2*sin(x).^2); %E(a,q)F=integral(fun1,0,a)E=integral(fun2,0,a)5、计算结果

歌曲名:Integral (Live At The Mermaid Theatre)歌手:Pet Shop Boys专辑:Concrete - In Concert At The Mermaid Theatre For Radio 2 With The Bbc Concert OrchestraIntegralPet Shop BoysIf you"ve done nothing wrongyou"ve got nothing to fearIf you"ve something to hideyou shouldn"t even be hereLong live usthe persuaded weintegralcollectivelyto the whole projectit"s brand newconceived solelyto protect youOne worldOne reasonUnchangingOne seasonIf you"ve done nothing wrongyou"ve got nothing to fearIf you"ve something to hideyou shouldn"t even be hereYou"ve had your chancenow we"ve got the mandateIf you"ve changed your mindI"m afraid it"s too lateWe"re concernedyou"re a threatYou"re not integralto the projectSterileImmaculateRationalPerfectEveryone hastheir own numberin the system thatwe operate underWe"re moving toa situationwhere your lives existas informationOne worldOne lifeOne chanceOne reasonAll underone skyunchangingone seasonIf you"ve done nothing wrongyou"ve got nothing to fearIf you"ve something to hideyou shouldn"t even be hereYou"ve had your chancenow we"ve got the mandateIf you"ve changed your mindI"m afraid it"s too lateWe"re concernedyou"re a threatYou"re not integralto the projectOne worldOne lifeOne chanceOne reasonAll underone skyunchangingone seasonIf you"ve done nothing wrongyou"ve got nothing to fearIf you"ve something to hideyou shouldn"t even be hereYou"ve had your chancenow we"ve got the mandateIf you"ve changed your mindI"m afraid it"s too lateWe"re concernedyou"re a threatIf you"ve done nothing wrongyou"ve got nothing to fearIf you"ve something to hideyou shouldn"t even be hereWe"re concernedyou"re a threatYou"re not integralto the projectSterileImmaculateRationalPerfectEnd-Xiao Wuhttp://music.baidu.com/song/7950913

int是符号积分integral是数值积分，在R2012版本以上才支持

你说的这个函数不是C语言的库函数吧下面的程序可能会帮助你#include <stdio.h>#include <math.h>double integral(double a, double b){ double s,h,y; int n,j; s=(sin(a)+sin(b))/2.0; n=100; h=(b-a)/n; for(j=0;j<n;j++) s=s+sin(a+j*h); y=s*h; return(y);}main( ){ double y; y=integral(0,1); printf("%f ",y);}

int 对符号函数进行积分,给出表达式（如果有的话）quad 数值积分,给出数值quad 只是一个参数可以是inline的函数,两者没太大联系

积分符号∫，来自于summa这个希腊单词的首字母，可以简化读作sum。当然，事实上很多人都读作“积分”，“……的积分”或“对……积分”，包括目前所有教过我的老师。外国读作“integral”，“theintegralof”等。所以，其实这个东西就是一个约定俗成的读法，怎么读都行。（可参考数学老师对各种希腊字母的各种方言读法:-D）

一般写成integral part 或者inte-gral part 意思是 1（不可缺少的）组成部分、主要部分、 Music is an integral part of the school"s curriculum. 音乐是这所学校的课程中基本的一环. 2整数部份（例如3.2 的3 就是integral part）

更新1: plz give some examples~~~ and it would be great if u can explain more clearly with the formulas~~ In 3 dimensions You can have o type of surface areas 1. it bounds a volume which me it is closed. To work out the Surface area(S. A.) you need to know the shape of the volume and all the planes on the volume e.g. a cube is bounded by 6 surfaces and you know the way to work out the area of each surface. e.g. a sphere. The formula is S.A. = 4*Pi*r^2 2. it is open and is called a curve(plane). In this case you need to know the regions of the S.A. If it is not a flat plan it is always hard to be calculated and here it es the Surface Integral Normally we express a surface as a function of x and y i.e. z=f(x y) We would like to express it as a vector function: r=xi+yj+zk and we want to have 2 variables instead of 3 so let x=u y=v then z=f(u v) and r bees r=ui+vj+f(u v)k (note that x=u y=v is not necessary it can be in other forms) then we differentiate r with respect to u and v and we get ru and rv |ru x rv| is a parallelogram This area approximates the area of a *** all part of the surface Adding these *** all areas(In the sense of the Riemann Integral) We get the area of the surface: A(s)=int(int(|ru x rv|)du)dv in a Domain which is the region of the surface Besides working out S.A. Surface Integral can work out the integration of a function on a plain. The formula will bee int(G(x y z))ds where G is the function we want to integrate and s is the surface we want to integrate on. It can be written as I = int(int( G( x(u v) y(u v) z(u v) )*|ru x rv| )du)dv I hope that is enough for you. It is quite hard for you to learn it on your own so I suggest you wait until you get a place in university. Good luck 参考: My lecture notes

定积分的数学符号就读“……的积分”。

1. (1) 不对(2) 不对，指令窗中显示的数值有效位数可以用format函数更改2.第一行如果把8后面的“;”改为“,”就正确第二行是正确的第三行使用的是中文标点，matlab会认为是错误

问题一：银行卡的“积分”，这里的“积分”，用英语该怎么说？ 给你问了一个在外企银行工作的朋友，她说bonus points或者reward points都可以，简写一般是BP。 问题二：“积分兑换”用英语怎么说 “积分兑换”用英语 credits exchange 问题三：会计中的积分制用英语怎么说 会计中的积分制 Integral system in accounting 会计中的积分制 Integral system in accounting 问题四：求导和积分用英语怎么说 Derivation and integration 例句： 仅用幂函数的求导和积分的形式运算公式和初等的方法求出了自然数有限项幂和当幂为非负整数时的求和公式。 This paper gives the formulas of the sums when the power is non-negative integersthrough the derivation and integration of the power function and by using theelementary method. 问题五：积分兑换商品的 “积分” 用英文怎么讲？ credit 问题六：信用卡积分英文怎么说？ point（s） of credit card信用卡积分 比如：If you join our membership, you can earn 1 point储member"s credit for each dollar you spend. 如果您成为会员，您每消费一美元,就会获得一分积分。 问题七：积分兑换英语怎么说？ Integral Exchange.

如何用计算机求解极其复杂的定积分？从题主给出的积分表达式，如用int积分函数求解是无法得到其解析值或数值解，可以考虑用数值积分函数来求解。如integral函数，其求解过程为：1、利用句柄函数来自定义积分表达式的函数，即fun=@(x) 自定义积分表达式2、确定被积函数的积分区间，如a(下区间)，b(上区间) 3、使用integral函数求其积分值，即Q = integral(fun,a,b)

bonus points或reward points均可，简写一般为BP。

用matlab求解含积分的方程组，可以这样实现：1、先用integral函数，求解定积分，即integral(@(t)sqrt(a^2.*sin(t).^2+b^2.*cos(t).^2),0,pi/2)2、再用vpasolve函数，求解β、θ值，即[beta1,theta1]=vpasolve(eq1,eq2, [beta1,theta1])这里：eq1是指方程一，eq2是指方程二3、详细的运行代码为a=15;b=14;r=14.5;syms theta1 beta1eq1=integral(@(t)sqrt(a^2.*sin(t).^2+b^2.*cos(t).^2),0,pi/2)+beta1*pi*r/180;eq2=(a*sin(theta1))/(b*cos(theta1));[beta1,theta1]=vpasolve(eq1==100,eq2==tan(beta1), [beta1,theta1])这里:beta1是指β，theta1是指θ运行结果

integral module estimate积分模估计u^ 2+ v^ 2 And, that is an integral that we know how to do.得到ab*dudv的二重积分，其积分区域为2，显然我们知道如何计算这种积分。

never7 ever17 remember11 12riven code1812riven是integral系列其他都是infinity系列系列巅峰是EVER17R11算是纯fans向，没有解答篇……code18未汉化另有DUNAMIS15，但是code18和D15都不是打越钢太郎的作品，所以没有汉化版也无所谓了，牌子只是次要，谁写的脚本才是最主要的

1、不定积分=indefiniteintegral　　不定积分，就是求一个被积函数integrand的原函数antiderivativefunction；　　一个函数f(x)求导后，得到导函数derivativefunction；　　把导函数当成被积函数，计算出原来的函数f(x)，f(x)就被称为原函数。　　2、定积分=definiteintegral　　在不考虑被积函数有间断点的情况下，定积分的方法，跟不定积分的方法一样；　　但是不定积分积不出来的情况，有很多在定积分的情况下就能积分出来,也就是说，不定积分，没有积分区间；定积分有积分区间，有时在特殊的积分区间上，不定积分无法积分，定积分却可以积出来。　　3、反常积分=improperintegral　　汉语中分成了两类：广义积分、暇积分。　　广义积分，就是涉及到积分区间，一侧或两侧出现无穷的情况；　　暇积分：就是积分区间中有间断点的积分。　　无论是广义积分，还是暇积分，积分方法与定积分没有差别,反常积分就是定积分，反常积分与一般的定积分的区别在于：积分后必须取极限才能得到结果。

如何用matlab求解带积分的二元方程组？分析题主给出的求解代码存在着下列几个方面的问题：问题一，由于给出的被积函数含有未知变量x1、x2，所以用integral函数不能求其积分值，而出现出错警告；解决思路，设置未知变量x1、x2分别为某具体值，然后再用integral函数求其积分值。问题二，在这里使用 vpasolve函数求解变量x1、x2不是太恰当；由于方程里含有不完整的积分式，所以用 vpasolve函数无法得到解。解决思路，使用fsolve函数求解其方程的数值解。解决步骤：1、创建求解带积分的二元方程组函数，f =func(x)。其内容：f1=@(t)exp(-(1-x2).*t)./t; %定义方程f2=@(t)exp(-x1*t)./t; %定义方程S1=integral(f1,1,+inf); %求积分值S2=integral(f2,1,+inf); %求积分值f = [x1-S1,x2-(1- S2)]; %2、用fsolve函数求解。即sol = fsolve(@func,rand(1,2));x1=sol(1) %x1的解x2=sol(2) %x2的解3、完善代码后运行可以得到x1 = 0.52761x2 = 0.47239

翻译还是自己来吧This article is about the concept of integrals in calculus. For the set of numbers, see integer. For other uses, see Integral (disambiguation).A definite integral of a function can be represented as the signed area of the region bounded by its graph.Topics in Calculus Fundamental theoremLimits of functionsContinuityMean value theorem [show]Differential calculus DerivativeChange of variablesImplicit differentiationTaylor"s theoremRelated ratesIdentitiesRules:Power rule, Product rule, Quotient rule, Chain rule[show]Integral calculus IntegralLists of integralsImproper integralsIntegration by:parts, disks, cylindricalshells, substitution,trigonometric substitution,partial fractions, changing order[show]Vector calculus GradientDivergenceCurlLaplacianGradient theoremGreen"s theoremStokes" theoremDivergence theorem [show]Multivariable calculus Matrix calculusPartial derivativeMultiple integralLine integralSurface integralVolume integralJacobian Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. Given a function u0192 of a real variable x and an interval [a, b] of the real line, the definite integralis defined informally to be the net signed area of the region in the xy-plane bounded by the graph of u0192, 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 u0192. 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 17th century. Through the fundamental theorem of calculus, which they independently developed, integration is connected with differentiation: if u0192 is a continuous real-valued function defined on a closed interval [a, b], then, once an antiderivative F of u0192 is known, the definite integral of u0192 over that interval is given byIntegrals and derivatives became the basic tools of calculus, with numerous applications in science and engineering. A rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. Beginning in the nineteenth century, more sophisticated notions of integrals began to appear, where the type of the function as well as the domain over which the integration is performed has been generalised. A line integral is defined for functions of two or three variables, and the interval of integration [a, b] is replaced by a certain curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space. Integrals of differential forms play a fundamental role in modern differential geometry. These generalizations of integral first arose from the needs of physics, and they play an important role in the formulation of many physical laws, notably those of electrodynamics. There are many modern concepts of integration, among these, the most common is based on the abstract mathematical theory known as Lebesgue integration, developed by Henri Lebesgue.Contents [hide]1 History 1.1 Pre-calculus integration 1.2 Newton and Leibniz 1.3 Formalizing integrals 1.4 Notation 2 Terminology and notation 3 Introduction 4 Formal definitions 4.1 Riemann integral 4.2 Lebesgue integral 4.3 Other integrals 5 Properties 5.1 Linearity 5.2 Inequalities for integrals 5.3 Conventions 6 Fundamental theorem of calculus 6.1 Statements of theorems 7 Extensions 7.1 Improper integrals 7.2 Multiple integration 7.3 Line integrals 7.4 Surface integrals 7.5 Integrals of differential forms 7.6 Summations 8 Methods 8.1 Computing integrals 8.2 Symbolic algorithms 8.3 Numerical quadrature 9 See also 10 Notes 11 References 12 External links 12.1 Online tools 12.2 Online books [edit] HistorySee also: History of calculus[edit] Pre-calculus integrationIntegration can be traced as far back as ancient Egypt ca. 1800 BC, with the Moscow Mathematical Papyrus demonstrating knowledge of a formula for the volume of a pyramidal frustum. The first documented systematic technique capable of determining integrals is the method of exhaustion of Eudoxus (ca. 370 BC), which sought to find areas and volumes by breaking them up into an infinite number of shapes for which the area or volume was known. This method was further developed and employed by Archimedes and used to calculate areas for parabolas and an approximation to the area of a circle. Similar methods were 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 the volume of a sphere.[1] That same century, the Indian mathematician Aryabhata used a similar method in order to find the volume of a cube.[2]The next major step in integral calculus came in Iraq when the 11th century mathematician Ibn al-Haytham (known as Alhazen in Europe) devised what is now known as "Alhazen"s problem", which leads to an equation of the fourth degree, in his Book of Optics. While solving this problem, he performed an integration in order to find the volume of a paraboloid. Using mathematical induction, he was able to generalize his result for the integrals of polynomials up to the fourth degree. He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree.[3] Some ideas of integral calculus are also found in the Siddhanta Shiromani, a 12th century astronomy text by Indian mathematician Bhāskara II.The next significant advances in integral calculus did not begin to appear until the 16th century. At this time the work of Cavalieri with his method of indivisibles, and work by Fermat, began to lay the foundations of modern calculus. Further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation.At around the same time, there was also a great deal of work being done by Japanese mathematicians, particularly by Seki Kōwa.[4] He made a number of contributions, namely in methods of determining areas of figures using integrals, extending the method of exhaustion.[edit] Newton and LeibnizThe major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Newton and Leibniz. The theorem demonstrates 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 Newton and Leibniz developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions within continuous domains. This framework eventually became modern calculus, whose notation for integrals is drawn directly from the work of Leibniz.[edit] Formalizing integralsWhile Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of rigour. Bishop Berkeley memorably attacked infinitesimals as "the ghosts of departed quantities". Calculus acquired a firmer footing with the development of limits and was given a suitable foundation by Cauchy in the first half of the 19th century. Integration was first rigorously formalized, using limits, by Riemann. Although all bounded piecewise continuous functions are Riemann integrable on a bounded interval, subsequently more general functions were considered, 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.[edit] NotationIsaac 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 or , which Newton used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted.The modern notation for the indefinite integral was introduced by Gottfried Leibniz in 1675 (Burton 1988, p. 359; Leibniz 1899, p. 154). He adapted the integral symbol, ∫, from an elongated letter s, standing for summa (Latin for "sum" or "total"). The modern notation for the definite integral, with limits above and below the integral sign, was first used by Joseph Fourier in Mémoires of the French Academy around 1819–20, reprinted in his book of 1822 (Cajori 1929, pp. 249–250; Fourier 1822, §231).