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# Integral Calculus

Integral Calculus

Integral Calculus

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### Integral Calculus

1. 1. INTEGRAL CALCULUS By- Manish Sahu M.Sc. Chemistry (Final) Sp.- Physical Chemistry
2. 2. SYNOPSIS  Introduction  History  Definition  Rules  Graphical Representations  Techniques of Integration  Application of Integration  Conclusion  Reference
3. 3. INTRODUCTION  The relationship involving the rate of change of two variables.  But also needed to know the direct relationship between of two variables.  For example, we may know the velocity of an object at a particular time, but we may went to know the position of the object at that time.  To find this direct relationship , we need to use the process which is opposite to differentiation . This is called INTEGRATION.
4. 4. HISTORY  Integration can be traced as far back as ancient Egypt before 1800 BC.  Further developed and employed by ARCHIMEDES and used to calculate areas for Parabolas .  Similar methods were independently developed in in china around the 3rd century AD by LIU HUI .  Next major step in INTEGRAL CALCULUS came in Iraq when the 11th century mathematician IBN AL- HAYTHAM ( Known as ALHAZEN in Europe ) .
5. 5.  Also formulated independently by ISAAC NEWTON and GOTTRIED LEIBNIZ in the late of 17th century .  Acquired a firmer footing with the development of limit and was given and a suitable foundation by CAUCHY in the first half of the 19th century .  INTEGRATION was first rigorously formalized ,using limits, by RIEMANN.  Other definitions of INTEGRAL, extending REIMANN’s and LEBESGUE’s approaches, were proposed. HISTORY
6. 6. DEFINITION This article is about the concept of definite integral in calculus . For the indefinite integral , see Antiderivatives.
7. 7. RULES  ∫ sinx dx = - cosx  ∫ cosx dx = sinx  ∫ tanx dx = - log(cosx)  ∫ cotx dx = log(sinx)  ∫ secx dx = log(secx +tanx )  ∫ cosecx dx = log(cosec – cotx )  ∫ x n dx = x n+1 / n+1  ∫ 1/x dx = logx  ∫ 1 dx = x  ∫ e x dx = e x  ∫ ax dx = ax / logea
8. 8. GRAPHICAL REPRESENTATIONS In mathematics ,an INTEGRAL assigns numbers to functions in a way that can describe displacement, area, volume , And other concepts that arise by combining infinitesimal data. INTEGRATION is one of the two main operations of calculus , with its inverse operation, Differentiation , being the other. Given a function f of a real variables x and an interval [a,b] of the real line ,the definite INTEGRAL is defined informally as the signed area of the region in the xy- plane that is bounded by the graph of f, the x – axis , ∫a b f (x) dx vertical lines x=a ,and x=b . The area above the x-axis adds to the total and that below the x-axis subtracts form the total.
9. 9. TECHNIQUES OF INTEGRATION Various techniques of integration ;  Integration by General Rule  Integration by Parts  Integration by Substitution
10. 10. Integration by General Rule ∫ xn dx = xn+1 / n+1 Example :- ∫ x4 dx ∫ x4 dx = x4+1 ⁄ 4+1 = x5 / 5
11. 11. Integration By Parts :- ∫ f(x).g(x) dx = f(x) ∫g(x)dx - ∫ {d/dx f(x) ∫g(x)dx } dx Example :- ∫ x.cosx dx ∫ x.cosx dx = x ∫ cosx dx - ∫ {d/dx(x) ∫ cosx dx } dx ∫ x.cosx dx = x ∫ cosx dx - ∫ { 1. sinx } dx = x ∫ cosx dx - ∫ sinx dx = xsinx – (- cosx ) = xsinx + cosx
12. 12. Integration by Substitution Example :- ∫ 4x 3 (x 4 +1) dx Let u = x4 +1, Then du = 4x3 dx ∫ 4x 3 (x 4 +1) dx = ∫ u du = u 2 /2 = ( x 4 +1 ) 2 /2
13. 13. APPLICATION OF INTEGRATION  The PETRONES TOWERS in Kuala Lumpur experience high forces due to winds. INTEGRATION was used to design the building for strength .  The SYDNEY OPERA HOUSE is very unusual design based on slices out of a ball . Many Differential equation were solved in the design of this Building .  Historically ,one of the first uses of INTEGRATION was in finding the volumes of wine casks .
14. 14. CONCLUSION  Very important Mathematical tool .  Used in many fields .  Important in Business .  Helps to estimate things like –  Marginal Cost  Marginal Revenue  Profit  Gross Loss
15. 15. REFERENCE  “Mathematics For Chemists” By :- “Bhupendra Singh”  “A History of the Definite Integral” By :- “Kallio & Bruce Victor”