Solved Problems on Numerical Integration. Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Review of the Subject.

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Presentation transcript:

Solved Problems on Numerical Integration

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Review of the Subject

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Definite Integrals

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä NUMERICAL APPROXIMATIONS Decompose [a,b] into n subintervals. Length of a subinterval: k th subinterval: Riemann sum: Tag-points t k can be chosen freely.

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Left Approx. Right Approx. APPROXIMATIONS FOR

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Midpoint Approximation MID(n) = Trapezoidal Approximation APPROXIMATIONS FOR

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä SIMPSONS APPROXIMATION In many cases, Simpsons Approximation gives best results.

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä PROPERTIES LEFT(n) If f is increasing, Property RIGHT(n)

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä COMPARING APPROXIMATIONS Property ab f If f is increasing and concave-up,

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problems

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problems 1 2 Speed given by table. Estimate the distance traveled. 3 t (s) s (m/s)

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä 4 Approximate the value of the integral Which method gives the best result? 5 Approximate the value of the integral Estimate the errors. Problems

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Integrals from graphs Problem Compute the integral Solution Draw the graph of the function and compute the integral as the area under the graph. First get rid of the absolute value signs.

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Solution INTEGRALS FROM GRAPHS

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Solution INTEGRALS FROM GRAPHS

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Solution INTEGRALS FROM GRAPHS

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Solution INTEGRALS FROM GRAPHS

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Solution y = ||x + 1| - |x - 1|| INTEGRALS FROM GRAPHS

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Solution The integral is the area of the yellow domain INTEGRALS FROM GRAPHS

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Solution Area = = INTEGRALS FROM GRAPHS

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Answer -221 INTEGRALS FROM GRAPHS

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Estimate using left Riemann sums with 12 subintervals of equal length. Problem INTEGRALS FROM GRAPHS

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Solution Division points: (-3,-2.5,-2,-1.5, -1,-0.5,0,0.5,1, 1.5,2,2.5,3). As tag points t k, use the left end-points. INTEGRALS FROM GRAPHS

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem tktk f(t k ) tktk f(t k ) INTEGRALS FROM GRAPHS

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Answer tktk f(t k ) tktk f(t k ) Left(12) estimate = 0.5( ) 11 INTEGRALS FROM GRAPHS

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Average Value of a Function Problem The average value of the function f on the interval [- 3,3] is 1.8.

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Functions given by tables Problem The speed of a racing car during the first 10 second of a race is given in the table below. Estimate the distance traveled during that time. t (s) s (m/s)

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Estimate the distance traveled m/s seconds FUNCTIONS GIVEN BY TABLES

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Estimate the distance traveled. t (s) s (m/s) Time intervals: 1 second, Δt = 1 (s). k v v = the average velocity during time interval. FUNCTIONS GIVEN BY TABLES

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Estimate the distance traveled. Time intervals: 1 second, Δt = 1 (s). k v (m/s) v = the average velocity during time interval. d = distance traveled during time interval. d (m) FUNCTIONS GIVEN BY TABLES

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Estimate the distance traveled. Time Speed Distance traveled = speed × time = total area of the rectangles FUNCTIONS GIVEN BY TABLES

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Estimate the distance traveled. k v (m/s) d (m) Distance traveled during 10 seconds = 403 m. Average speed 40.3 m/s 90 mph. FUNCTIONS GIVEN BY TABLES

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Estimate the distance traveled. Time Speed s(t) = speed of an object at time t. Distance traveled during time interval [a,b] DISTANCE AS AN INTEGRAL OF SPEED

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Time Speed FORMULA 1 RACE CAR Acceleration 0 to 200 km/h (124 mph): 3.8 s. Deceleration: up to 5-6 g (48-58 m/s 2 ).

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Approximate the value of the integral Which method gives the best result? COMPARING METHODS Problem

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Approximate COMPARING METHODS Solution The integral is easy to compute:

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä COMPARING METHODS Solution For this integral: LEFT(1) = 0MID(1) =1/8RIGHT(1) = 1 TRAP(1) = 1/2 RIGHT(1) = 1 Problem Approximate

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä COMPARING METHODS Solution For this integral: RIGHT(1) = 1 SIMPSON(1)= Problem Approximate

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä COMPARING METHODS Conclude Simpsons Approximation gives the precise value of the integral. RIGHT(1) = 1 This is true for integrals of polynomials of degree at most 3. Problem Approximate

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Approximate the integral Estimate the errors. INTEGRALS OF BELL SHAPED CURVES -2 2

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Estimate the errors. INTEGRALS OF BELL SHAPED CURVES Solution The function is decreasing for 0 x 1. Hence for all n.

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Estimate the errors. INTEGRALS OF BELL SHAPED CURVES Solution Computing with a computer we get Error <

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Estimate the errors. INTEGRALS OF BELL SHAPED CURVES Solution Observe that

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Estimate the errors. INTEGRALS OF BELL SHAPED CURVES Solution Hence the graph of f is concave down for -1 1 or x < -1. Hence

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Estimate the errors. INTEGRALS OF BELL SHAPED CURVES Solution Likewise This yields (with n = 10):

Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Estimate the errors. INTEGRALS OF BELL SHAPED CURVES Solution We get Computation with a computer algebra system, yields the more accurate estimate: