Chapter17 - Problem 17.3. /2 Given the integral (6 3cos...

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Unformatted text preview: Problem 17.3. /2 Given the integral (6 3cos x)dx (6 x 3sin x) | 0 /2 0 3 3 12.4247 Evaluate it using 4 equally spaced points in the integration interval We first generate the data >> f=@(x) 6+3*cos(x) f = @(x)6+3*cos(x) >> x=linspace(0,pi/2,4) x = 0 0.5236 1.0472 1.5708 >> y=f(x) y = 9.0000 8.5981 7.5000 6.0000 We can first integrate using trapezoidal integration >> trapez=(pi/12)*(y(1)+2*y(2)+2*y(3)+y(4)) trapez = 12.3559 Or >> trapez=trapz(x,y) trapez = 12.3559 For an error of 0.0688 Or we can use Simpson rule on the first two points, and trapezoidal rule on the last two. The interval for Simpson rule is 2/3 of the original interval or pi/3. >> Simp=(pi/18)*(y(1)+4*y(2)+y(3))+(pi/12)*(y(3)+y(4)) Simp = 12.4167 With an error of 0.008. Or we can fit the data with a polynomial or a spline, and then use a very dense grid to integrate with trapezoidal rule. >> xx=linspace(0,pi/2); >> yy=polyval(p,xx); >> trapezfit=trapz(xx,yy) trapezfit = 12.4277 With an error of 0.004. We can check if the error is due to the fit or the limited number (100) of points by doubling the number of points. >> xx=linspace(0,pi/2,200); >> yy=polyval(p,xx); >> trapezfit=trapz(xx,yy) trapezfit = 12.4278 So the error is due to the fit. For this problem, we get the same result with other fits: >> yy1=interp1(x,y,xx,'cubic'); >> trapezcub=trapz(xx,yy1) trapezcub = 12.4278 >> yy1=interp1(x,y,xx,'spline'); >> trapezcub=trapz(xx,yy1) trapezcub = 12.4278 ...
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