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Solved Problems on Numerical Integration

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

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

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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.

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

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

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

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

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

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

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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)012345678910 s (m/s)0142842556730 455770

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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

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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.

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

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

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

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

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

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Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Solution The integral is the area of the yellow domain. -221 INTEGRALS FROM GRAPHS

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Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Solution Area = 2 + 1 + 1 + 2 = 6. -221 INTEGRALS FROM GRAPHS

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

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Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä 123-3-2 2 3 1 Estimate using left Riemann sums with 12 subintervals of equal length. Problem INTEGRALS FROM GRAPHS

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Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem 123-3-2 1 2 3 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

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Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä 123-3-2 2 3 1 Problem tktk f(t k ) 0.52.12 12.68 1.52.75 22.48 2.51.98 31.25 tktk f(t k ) -31.6 -2.51.76 -21.75 -1.51.37 1.0 -0.51.0 01.5 INTEGRALS FROM GRAPHS

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Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Answer tktk f(t k ) 0.52.12 12.68 1.52.75 22.48 2.51.98 31.25 tktk f(t k ) -31.6 -2.51.76 -21.75 -1.51.37 1.0 -0.51.0 01.5 Left(12) estimate = 0.5(1.6 + 1.76 + 1.75 + 1.37 + 1.0 + 1.0 + 1.5 + 2.12 + 2.68 + 2.75 + 2.48 + 1.98) 11 INTEGRALS FROM GRAPHS

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Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Average Value of a Function 123-3-2 2 3 1 Problem The average value of the function f on the interval [- 3,3] is 1.8.

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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)012345678910 s (m/s)0142842556730 455770

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Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Estimate the distance traveled. 1 2 3 4 5 6 7 12345678910 10 m/s seconds FUNCTIONS GIVEN BY TABLES

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Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Estimate the distance traveled. t (s)012345678910 s (m/s)0142842556730 455770 Time intervals: 1 second, Δt = 1 (s). k12345678910 v7213548.56148.53037.55163.5 v = the average velocity during time interval. FUNCTIONS GIVEN BY TABLES

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Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Estimate the distance traveled. Time intervals: 1 second, Δt = 1 (s). k12345678910 v (m/s)7213548.56148.53037.56163.5 v = the average velocity during time interval. d = distance traveled during time interval. d (m)7213548.56148.53037.55163.5 FUNCTIONS GIVEN BY TABLES

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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

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Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Problem Estimate the distance traveled. k12345678910 v (m/s)7213548.56148.53037.56163.5 d (m)7213548.56148.53037.55163.5 Distance traveled during 10 seconds = 403 m. Average speed 40.3 m/s 90 mph. FUNCTIONS GIVEN BY TABLES

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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

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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 ).

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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

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

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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

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

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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

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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

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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.

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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 < 0.173.

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

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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

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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):

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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:

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Section 8.5 Riemann Sums and the Definite Integral.

Section 8.5 Riemann Sums and the Definite Integral.

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