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  1. 1. Engineering Mathematics- Integration
  2. 2. LEARNING OBJECTIVES• To understand the virtual loss of GM and the calculations.• To calculate the maximum trim allowed to maintain a minimum stated GM.• To understand the safe requirements for a ship prior enter into dry dock. To understand the critical period during dry docking process.
  3. 3. Course Outline• Name of Course :Chief and Second Engineer 3000 kW or more (Unlimited Voyage)• Course Code/Module : ECSU , Part A• Subject : Mathematics and Engineering Drawing
  4. 4. Course OutlineModule Aims• To provide students with the familiarization to the fundamentals of calculus Mathematics required for engineering practice and problem solving.General Learning Objective - GLO• Recognize that integration is the inverse process of differentiation, and apply this knowledge to determine the area/volume/work done.Specific Learning Objectives - SLO Recognize that integration can be considered the reverse of differentiation process. Explain the integration of x, trigo. functions, 1/x, exponential functions. Evaluate the constant of integration. Perform the definite Integral.Apply integration to find:• a. Area under curves.• Volume of solid revolution• Work done• Mean & root mean square (rms) values• Centroid
  5. 5. Course outline• Instructional Hours• Lecture : 40 hours• Topics Hours• Integration as reverse of differentiation 2• Integration of functions: x,Trig,1/x, Exponential 8• Evaluation of constant of integration 4• Definite integral 6• Application of integral calculus to: 20a. Area under curves.b. Volume of solid revolutionc. Work doned. Mean & root mean square (rms)e. valuesCentroid20
  6. 6. Course OutlineIntegration as the Process of SummationIntegration as the Reverse of DifferentiationIntegration of functionsApplications of Integration : Areas Bounded by Curves and Volumes of Revolution
  7. 7. • Teaching Methods -• Combination of combination of methods as necessary - lectures, practice• Assessment Methods• Lecturer Class Assessment 1 20 %• Lecturer Class Assessment 2 20 %• Lecturer Class Assessment 3 20 %• Final Exam 40 %• Recommended Texts• K A Stroud (1992), Engineering Mathematics Programmes And Problems• G.S.Sharma & I.J.S.Sarna (1992), Engineering Mathematics
  8. 8. Integration : Concept and Theory We know how to find the area of simple geometric shapes such as the triangle below y 2 1 x 1 2
  9. 9. Integration : Concept and Theory But how do we find the are of geometric object which do not have straight edges ? y f(x) x a b
  10. 10. Integration : Concept and Theory So, how do we go about finding the area under the curve f(x), between x=a and x=b ? Well,  we can divide the area under the curve into separate rectangles …  … find the area of each rectangle …  … and then sum these areas in order to find an approximate answer to area under curve
  11. 11. Integration : Concept and Theory y f(x) x h a b Find area of each rectangle … … then sum all areas between x=a and x=b
  12. 12. Process of Integration • Integration is reverse of differentiation • In differentiation, if f(x)= then f`(x)= 4x . Thus the integral of • integration is the process of moving from f`(x) to f(x). By similar reasoning, the integral of. • Integration is a process of summation or adding parts together and an elongated S, shown as, is used to replace the words ‘the integral of’. Hence, from above,‘c’ is called the arbitrary constant of integration
  13. 13. Integration is the reverse process of differentiation. - (PNO2)-n dPNO2 = k dt Power series integration,increase the exponent by one - (PNO2)-n+1 kt = +C -n+1 1 Constant and divide by the of integration new exponent.
  14. 14. The Fundamental Theorem of Calculus b If F x   f x , then a f  xdx  Fb  Fa .If we know an anti-derivative, we can use it to find thevalue of the definite integral.
  15. 15. The Fundamental Theorem of Calculus b If F x   f x , then a f  xdx  Fb  Fa .If we know an anti-derivative, we can use it to find thevalue of the definite integral.If we know the value of the definite integral, we can use itto find the change in the value of the anti-derivative.
  16. 16. Integration of function • The general solution of integrals of the form • axndx, where a and n are constants is given by: x n1  x dx  n  1  C (n  1). nThis rule is true when n is fractional, zero, or apositive or negative integer, with the exception of n = -1.
  17. 17. Standard Integrals
  18. 18. • The integral of constant K is kx+c. for example Integral of several terms = Sum of integral of the separate terms for exemple
  19. 19. Problems
  20. 20. Exercise