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The Riemann Integral Anti-derivative Function and Indefinite Integrals Change of variables, Substitution Method Riemann Sums, Area and the Definite Integral The Fundamental Theorem of Calculus Numerical Integration The Riemann Integral Mongi BLEL Department of Mathematics King Saud University 2017-2018 Mongi BLEL The Riemann Integral Anti-derivative Function and Indefinite Integrals Change of variables, Substitution Method Riemann Sums, Area and the Definite Integral The Fundamental Theorem of Calculus Numerical Integration Table of contents 1 Anti-derivative Function and Indefinite Integrals 2 Change of variables, Substitution Method 3 Riemann Sums, Area and the Definite Integral 4 The Fundamental Theorem of Calculus 5 Numerical Integration Mongi BLEL The Riemann Integral Anti-derivative Function and Indefinite Integrals Change of variables, Substitution Method Riemann Sums, Area and the Definite Integral The Fundamental Theorem of Calculus Numerical Integration Indefinite Integrals & Anti-derivative Function Definition Let f : I −! R be a function defined on an interval I . A function F : I −! R is called an anti-derivative of f on I if F is differentiable on I and F 0(x) = f (x), forallx 2 I : There are many anti-derivatives of the function f (x) = 2x on R such as 11 F (x) = x2+1; F (x) = x2−7; F (x) = x2+ ; F (x) = x2+c: 1 2 3 2 4 Mongi BLEL The Riemann Integral Anti-derivative Function and Indefinite Integrals Change of variables, Substitution Method Riemann Sums, Area and the Definite Integral The Fundamental Theorem of Calculus Numerical Integration Proposition Let F and G be two anti-derivatives of a function f on an interval I , then there is a constant c 2 R such that F (x) = G(x) + c; 8x 2 I : Mongi BLEL The Riemann Integral Anti-derivative Function and Indefinite Integrals Change of variables, Substitution Method Riemann Sums, Area and the Definite Integral The Fundamental Theorem of Calculus Numerical Integration Definition Let F be an anti-derivative of a function f on an interval I , we Z denote f (x)dx any anti-derivative i.e. Z f (x)dx = F (x) + c; 8x 2 I (1) Z f (x)dx is called the indefinite integral of f on I . In the equation (1), • the constant c is called the constant of integration, • x is called the variable of integration, • f (x) is called the integrand. Z The mapping f 7−! f (x)dx is called as indefinite integral or evaluating the integral or integrating f . Mongi BLEL The Riemann Integral Anti-derivative Function and Indefinite Integrals Change of variables, Substitution Method Riemann Sums, Area and the Definite Integral The Fundamental Theorem of Calculus Numerical Integration Basic table of indefinite integrals Z f (x) f (x)dx xr+1 xr ; r 6= −1; R 2 + c Q r + 1 cos(x) sin(x) + c sin(x) − cos(x) + c sec2(x) tan(x) + c csc2(x) − cot(x) + c sec(x) tan(xMongi) BLEL The Riemannsec(x Integral) + c Anti-derivative Function and Indefinite Integrals Change of variables, Substitution Method Riemann Sums, Area and the Definite Integral The Fundamental Theorem of Calculus Numerical Integration Proposition (Some important formulas) • If f is differentiable on an interval I , then Z d f (x)dx = f (x) + c: dx • If f has an anti-derivative on an interval I , then d Z f (x)dx = f (x): dx Z Z αf (x)dx = α f (x)dx: Z Z Z f (x) + g(x)dx = f (x)dx + g(x)dx: Mongi BLEL The Riemann Integral Anti-derivative Function and Indefinite Integrals Change of variables, Substitution Method Riemann Sums, Area and the Definite Integral The Fundamental Theorem of Calculus Numerical Integration Substitution Method Theorem (Substitution) If F is an anti-derivative of f , then f (g(x))g 0(x) has anti-derivative F (g(x)). Or, Z f (g(x))g 0(x)dx = F (g(x)) + c: This is obvious. It is called "substitution" since it can be obtained by substituting u = g(x) and du = g 0(x)dx into Z f (u)du = F (u) + c: Mongi BLEL The Riemann Integral Anti-derivative Function and Indefinite Integrals Change of variables, Substitution Method Riemann Sums, Area and the Definite Integral The Fundamental Theorem of Calculus Numerical Integration Remark Substitution method is also called changing variable method. Examples Z Z n+1 2 n+1 2 n u=x2+1 n u (x + 1) 1 (x + 1) 2xdx = u du = = + c. n + 1 n + 1 Z u=2x+3 1 Z 1 2 sin(2x + 3)dx = sin udu = − cos u + c = 2 2 1 − cos(2x + 3) + c. 2 Z Z 1 u=πx 1 1 1 3 dx = du = tan(πx) + c. cos2(πx) π cos2(u) π Mongi BLEL The Riemann Integral Anti-derivative Function and Indefinite Integrals Change of variables, Substitution Method Riemann Sums, Area and the Definite Integral The Fundamental Theorem of Calculus Numerical Integration Theorem Let I be an interval, r 2 Q n {−1g and f : I ! R a differentiable function. Assume that f r (x) is defined for every x 2 I . Then Z f r+1(x) f r (x)f 0(x)dx = + c: r + 1 Mongi BLEL The Riemann Integral Anti-derivative Function and Indefinite Integrals Change of variables, Substitution Method Riemann Sums, Area and the Definite Integral The Fundamental Theorem of Calculus Numerical Integration Summation Notations Definition n X Given a set of numbers fa1; a2; :::; ang, the symbol ak k=1 represents their sum as follows n X ak = a1 + a2 + ::: + an: k=1 Mongi BLEL The Riemann Integral Anti-derivative Function and Indefinite Integrals Change of variables, Substitution Method Riemann Sums, Area and the Definite Integral The Fundamental Theorem of Calculus Numerical Integration Theorem For every c 2 R and n 2 N, we have n X c = nc: k=1 Theorem Let α; β 2 R and n 2 N. For every a1; a2; :::; an; b1; b2; :::; bn 2 R we have n n n X X X (αak + βbk ) = α ak + β bk : k=1 k=1 k=1 Mongi BLEL The Riemann Integral Anti-derivative Function and Indefinite Integrals Change of variables, Substitution Method Riemann Sums, Area and the Definite Integral The Fundamental Theorem of Calculus Numerical Integration Theorem For all n 2 N, we have n X n(n + 1) k = 2 k=1 n X n(n + 1)(2n + 1) k2 = 6 k=1 n X hn(n + 1)i2 k3 = : 2 k=1 Mongi BLEL The Riemann Integral Anti-derivative Function and Indefinite Integrals Change of variables, Substitution Method Riemann Sums, Area and the Definite Integral The Fundamental Theorem of Calculus Numerical Integration Example 1 : n n n n X X X X (3k2 − 2k + 1) = 3 k2 − 2 k + 1; k=1 k=1 k=1 k=1 n(n + 1)(2n + 1) n(n + 1) = 3: − 2: + 1:n; 6 2 nh i = (n + 1)(2n + 1) − 2(n + 1) + 2 ; 2 n = (2n2 + 3n + 1 − 2n − 2 + 2); 2 n = (2n2 + n + 1): 2 Mongi BLEL The Riemann Integral Anti-derivative Function and Indefinite Integrals Change of variables, Substitution Method Riemann Sums, Area and the Definite Integral The Fundamental Theorem of Calculus Numerical Integration Riemann Sums, Area and the Definite Integral The approach of the integral of function by areas gives the geometrical sense of integration. The second approach consists in introducing a priori the anti-derivative of function. The idea of the first approach is to cut the interval [a; b] by a subdivision in sub-intervals [aj ; aj+1], then to add the areas of rectangles based on the intervals [aj ; aj+1]. Mongi BLEL The Riemann Integral Anti-derivative Function and Indefinite Integrals Change of variables, Substitution Method Riemann Sums, Area and the Definite Integral The Fundamental Theorem of Calculus Numerical Integration Definitions 1 A partition P of the closed interval [a; b] is a finite set of points P = fa0; a1; a2;:::; ang such that a = a0 < a1 < a2 < : : : < an−1 < an = b. Each [aj−1; aj ] is called a subinterval of the partition and the number hj = aj − aj−1 is called the amplitude of this interval. 2 The norm of a partition is defined to be the length of the longest subinterval [aj ; aj+1], that is, it is jjPjj = maxfaj − aj−1; j = 1;:::; ng. 3 A partition P = fa0; a1; a2;:::; ang of the closed interval b − a [a; b] is a called uniform if a − a = . Then in this k+1 k n case b − a Mongia BLEL= a +Thek Riemann Integral: k n Anti-derivative Function and Indefinite Integrals Change of variables, Substitution Method Riemann Sums, Area and the Definite Integral The Fundamental Theorem of Calculus Numerical Integration Definitions 1 A mark on the partition P = fa0; a1; a2;:::; ang is a set of points w = fx1;:::; xng such that xj 2 [aj−1; aj ]. 2 A pointed partition of an interval is a partition of an interval together with a finite sequence of numbers x1; x2;:::; xn such that 8j = 1;:::; n; xj 2 [aj−1; aj ]: This pointed partition will be denoted by P = f([aj−1; aj ]; xj )g1≤j≤n: Mongi BLEL The Riemann Integral Anti-derivative Function and Indefinite Integrals Change of variables, Substitution Method Riemann Sums, Area and the Definite Integral The Fundamental Theorem of Calculus Numerical Integration Definition Let P = f([aj−1; aj ]; xj )g1≤j≤n be a pointed partition of the interval [a; b]. The Riemann sum of f with respect to the pointed partition P is the number n n X X R(f ; P) = f (xj )(aj − aj−1) = f (xj )∆j (2) j=1 j=1 Each term in the sum is the product of the value of the function at a given point and the length of an interval.
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