CPS 101 Introduction to Computational ScienceWensheng ShenDepartment of Computational ScienceSUNY BrockportIntegration is the reverse of differentiation. In previous chapter, we find the rate of change by giving change; here we find the total change by giving the rate of change.Chapter 9: Fundamental Concepts of Integral CalculusExample: constant rate of changeA car travels on an inter-state highway for 2 hours at a constant velocity of 65 miles/hour, how long does the car travel?A car travels on a state highway for 2 hours. It goes at a speed of 50 miles for the first half an hour, 45 miles for the second half an hour, 55 miles for the third half an hour, and 55 for the fourth half an hour. What is the total number of miles the car has traveled? dy/dt = constantFundamental theorem of calculus( )∫∆+∆+∆=−∞>−bannxxfxxfxxfdxxf )()()()(110limL∫−=baaFbFdxxf )()()(∫+= CxFdxxf )()(Example: variable rate of changedy/dt = -t2+ 10t + 24Velocity = -t2+ 10t + 24Numerical integrationGoal to find the integration of We use points a=x0,x1,x2,…,xn=b to divide an interval [a, b] into n equal-sized sub-intervalsWe calculate the y=f(x) at each points x0, x1, x2, …, xnas y0, y1, y2, …, yn. We compute the integration using special rules∫badxxf )(Rectangular rule)...()(21 nbayyynabdxxf +++−≈∫Trapezoid rule+++++−≈−∫1210...)(21)(nnbayyyyynabdxxfSimpson’s rule[ ])()(4)(2...)(4)(2)(4)(3)(123210 nnnbaxfxfxfxfxfxfxfhdxxf +++++++=−−∫[