Page - 88 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

88 3 Computing Integrals print ’Exact integral:’, I_exact.evalf() x0 = -2; x1 = 2; y0 = -2; y1 = 2 n = 1000 np.random.seed(6) I_expected = 16.7970837117376384 # Computed with this seed I_computed = MonteCarlo_double( lambda x, y: np.sqrt(x**2 + y**2), g, x0, x1, y0, y1, n) print ’MC approximation %d samples): %.16f’ % (n**2, I_computed) assert abs(I_expected - I_computed) < 1E-15 (See thefileMC_double.py.) 3.8 Exercises Exercise3.1:Handcalculations for the trapezoidalmethod Compute by hand the area composed of two trapezoids (of equal width) that ap- proximates the integral R3 1 2x3dx.Makea test functionthatcalls thetrapezoidal function in trapezoidal.py and compares the return value with the hand- calculatedvalue. Filename:trapezoidal_test_func.py. Exercise3.2:Handcalculations for themidpointmethod Compute by hand the area composed of two rectangles (of equal width) that ap- proximates the integral R3 1 2x3dx. Make a test function that calls the midpoint function inmidpoint.pyand compares the return valuewith the hand-calculated value. Filename:midpoint_test_func.py. Exercise3.3:Computeasimple integral Applythetrapezoidalandmidpointfunctions tocompute the integral R6 2 x.x 1/dxwith2and100subintervals. Compute theerror too. Filename:integrate_parabola.py. Exercise3.4:Hand-calculationswithsine integrals Weconsider integrating thesine function: Rb 0 sin.x/dx. a) Let b D and use two intervals in the trapezoidal and midpoint method. Compute the integral by hand and illustrate how the two numerical methods approximates the integral. Comparewith theexactvalue. b) Doa)whenbD2 . Filename:integrate_sine.pdf. Exercise3.5:Maketest functions for themidpointmethod Modify thefiletest_trapezoidal.pysuch that the three tests are applied to the functionmidpoint implementing themidpointmethodfor integration. Filename:test_midpoint.py.