Primitive indefinite integral. Definition Antiderivative The function F(x) is an antiderivative of the function f(x) on (a, b) if F(x) = f(x) ∀x ∈ (a, b). Let F(x) and G(x) be two antiderivatives of the function f(x) on (a, b), then there exists a constant C such that: F(x) = G(x) + C Indefinite integral The set of antiderivatives of f (x), recorded as ∫f(x)dx is called the indefinite integral of the function f(x). If F (x) is an antiderivative of f (x) then we have
Group Group Group Mathematical Analysis I Quỳnh Anh Dũng Hồng Mai Quỳnh 16 Thảo Mạnh Vũ Ngọc Group MATHEMATICAL ANALYSIS I Integral and its application About the introduction The overview about and development of the antiderivative theory of integration - integrals Content Ghi Nội dung Content Content Content Application of definite Conclusion and integrals references Group MATHEMATICAL ANALYSIS I Integral and its application Content The overview about antiderivative - integral 1.Antiderivative indefinite integral Content Ghi Nội dung Content Content Definite integral Integration method Group MATHEMATICAL ANALYSIS I Integral and its application Content Ghi Nội dung Content Some details on the birth and development of integral theory History of integrals A brief overview of the application of integrals Content Ghi Nội dung Content Group MATHEMATICAL ANALYSIS I Integral and its application Content Ghi Nội dung definite integrals Content Ghi Nội dung 2.Algebraic application of definite integrals Content Application of definite integrals Ghi Nội dung Content 1.Geometrical applications of Group Name MATHEMATICAL ANALYSIS I Integral and its application Content Ghi Nội dung Content Ghi Nội dung Conclusion Content Ghi Nội dung Content Conclusion CONTENT The overview about antiderivative - integrals 2.Area of flat figure NỘI DUNG Content A Area under the curves in Cartesian coordinates Apply the sum of integrals method to calculate the area of the plane figure S formed by two any functions f 1(x) and f2(x) that y can be divided into an infinite number of curved trapeziums MPFE y+∆y P y (Figure 1) It is easy to prove the following cases: M F E a O If the function y = f(x) is continuous on [a, b] then the area S under: x x +∆x b x y (C): y = f(x) +∆x a O b x NỘI DUNG Content Example 2: Calculate the area S under the curves: We have: S= S= = = Content If two functions y = f(x) and y = g(x) are continuous on [a, b] then the area S under the curves: NỘI DUNG Content NỘI DUNG Note: Four lines of the graph formed by two continuous functions f(x) and g(x) and two lines �= �; = b always create a plane However, just two graph of two continuous functions f(x) and g(x) are enough to form a plane figure, for example as shown in the following figure: Problem: Find the area of a plane figure S bounded by are two open curves and intersect to each other + Step 1: Solve the equation + Step 2: Use the formula: NỘI DUNG Content Example 3: Calculate the area under the curves: We consider the difference of two functions: Note: Equation (4) is only used to calculate the area of a limited simple plane (H) by the graph of the function f(x), g(x) and two lines x = a, x = b.In the case of Because at points whose coordinates are -2;0;1 So the area of the figure is: a plane figure (H) is defined by more than two graphs of the function, then to calculate the area (H), we must draw figure (H) – ie draw defined lines (H) Based on that, we divide the plane figure (H) into simple shapes simple that we know how to calculate Content Example 4: Calculate the area S of the plane figure (H) under by the graph of the function and straight line Equation of the intersection point: 1, 2, 3, We can easily see the area of the plane figure (H): (H) = = NỘI DUNG Content NỘI DUNG Area of a plane bounded by a line with parametric equations If the curve (C): y = f(x) has parametric equation Example : Calculate the area of an ellipse formed by (E) In the formula to calculate the area I replace We have the parametric equation of (E) on the first quadrant on the plane (Oxy) + y = f(x) by y = ω(t) ′ + dx by φ (t)dt + Two limits a and b are replaced by α and β which are solutions of a = φ(α) and b = φ(β) respectively Then : S= Converting the integral we get: If the curve (C) is closed, counter-clockwise and the area S is limited to the left and (C) has a parametric equation with ≤ t ≤ T where T is the period of it So Content 3 NỘI DUNG Arc length A Theoretical basis .We can subdivide this curve into an infinite number of “nearly-straight” lines Δl and sum them together Consider and Δx > such that Δx .With Δx small enough, we consider the length of the graph curve f(x) limiting between the two lines x= and x= is the length of the line connecting points A and B as shown in figure And also because Δx is small, so AB = (d) should be considered as the tangent at x of f(x) So the arc length Δl AB = , with a is the angle formed by the tangent AB at x of f(x) and Therefore: Δl=Δx (5) the axis Ox, so tanα=f’() NỘI DUNG Content 3 Acr length B Curve length in Cartesian coordinates From (5) we take the sum of the lengths of the small “straight” segments together, we get the formula to calculate the curve length limited by is L Then: L= Example 7: Calculate the length of the arc OA lying on the parapol y= with a≠0, which O is the origin, A is a point on the parabol whose latitude is t We have: A(t,) and y’(x) = Therefore L = = = = [x+ ] =+ Content 3 Acr length NỘI DUNG C Curve length with parametric equation We demonstrate the complete howarc to convert the area of a plane figure in Cartesian coordinates to the area of a plane bounded by a line with parametric equations (in section Example 8: Calculate the analogy lengthfrom of the of Archimedes’spiral 3.1.2) We have: x’(t) = -asint, y’(t) = acost, z’(t) = a From then, instead of (5) we get: l = dx in Therefore L = =dx in = =a =a=a= Content II NỘI DUNG ALGEBRAIC APPLICATIONS OF DEFINITE INTEGRAL applications Integral Integral inequality prove the inequality Example 12: Prove that: ln(1+a) > , for all a > We evalute the inequality in terms of functions and integrals +) In the coordinate system Oxy, consider the graph of the function (C): y = and set ϵ [1;a] Example 11: Prove that: < < Let the points A, B, C, D be A(1; 0), B(1+a; 0), C(; 0), D() Consider f(x) = is continuous on [0;] We have: y’ = => y’’ = = > 0, for all x > We have ≤ cosx ≤ 1, for all x ϵ [0;] Then: y = is a concave function, for all x > Then: ≤ ≤ 1, for all x ϵ [0;] Two lines x = 1, x = 1+a intersect the tangent (d) => at , for ϵ [0;] D()all at xtwo points E, F and intersect the graph (C): y = at two points H, G Content Conclusion Content Content Conclusion Essay content"Integral Applications”Including methods of calculating primitives, definite integrals and some applications of definite integrals, the essay has achieved some important results: + The essay has demarcated and presented the method of calculating the area of a plane figure, the length arc in geometrical applications of integrals From there, apply integration to practical problems and solve some common problems such as proving inequality + Is aimed at raising the reader's awareness of the integral and its application in mathematics Through the above analysis, we will have an overview and a more positive view of integrals, about applications that are very close to reality But in any field, two methods total integral and differential diagram are always two directions for easy access and use in practical problems Today, calculus - integration requires learners and researchers to have a deep understanding of the nature so that they can come up with new and more practical applications, not just stopping at things that the group has already learned tell In the process of implementation, the team has tried to limit errors but still cannot avoid mistakes, so the essay is still incomplete and profound The student group hopes to receive more comments from readers Finally, the group would like to express their sincere thanks to Mr Nguyen Van Thin's support during the process of writing the essay Thanks For Watching! Group ... definite Conclusion and integrals references Group MATHEMATICAL ANALYSIS I Integral and its application Content The overview about antiderivative - integral 1.Antiderivative indefinite integral Content... Definite integral Integration method Group MATHEMATICAL ANALYSIS I Integral and its application Content Ghi Nội dung Content Some details on the birth and development of integral theory History of integrals... the application of integrals Content Ghi Nội dung Content Group MATHEMATICAL ANALYSIS I Integral and its application Content Ghi Nội dung definite integrals Content Ghi Nội dung 2.Algebraic application