restart; Numerical Integration

In class we are learning various techniques of integration to find exact values of integrals. However, there are many situations where it is impossible to find the exact value of a definite integral. For instance, it may be impossible to find an antiderivative, or we may have observed data that does not have a known function associated with it. For these situations, there are techniques (often known as quadrature) that give numerical estimates to the integral. Maple can do this automatically. For example: int( exp(sin(x)), x=0..Pi );

LUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQtSSRleHBHRiQ2Iy1JJHNpbkdGJDYjSSJ4R0YnL0YvOyIiIUkjUGlHRiU=

evalf(%);

JCIrTyFlKDNpISIq

In this lab we consider some of the fundamentals of how this is done, although the simple methods we study here can't compete with the pros.

You have already learned about our first numerical integration method, Riemann sums, in Calculus I. Recall that 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and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUklbXN1cEdGJDYlLUYjNiMtRi82JVEiaUYnRjJGNS1GIzYjLUYvNiVRIUYnRjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRiM2Iy1JI21vR0YkNjBRIipGJy9GNlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlQvJSlzdHJldGNoeUdGVC8lKnN5bW1ldHJpY0dGVC8lKGxhcmdlb3BHRlQvJS5tb3ZhYmxlbGltaXRzR0ZULyUnYWNjZW50R0ZULyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS50aGlubWF0aHNwYWNlRicvJSdyc3BhY2VHRmBvLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnRkcvJS9zdWJzY3JpcHRzaGlmdEdGSS9JK21zZW1hbnRpY3NHRiRRLFtub25lLG5vbmVdRidGRA== is any point in the interval 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. (In fact, we defined the integral as the limit of those sums as n goes to infinity.) We often find it convenient to choose the "sampling point" 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either the left endpoint of the 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, the right endpoint of the interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYkLUYjNiMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtRjQ2JVEiaUYnRjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y7USdub3JtYWxGJw==, or the midpoint of the interval 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 To illustrate Riemann sums using the left endpoints, right endpoints, and midpoints for f(x) = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRic= we can use the following from the student package. with(student): with(plots): Lplot:=leftbox(x^2, x=1..3, 10, xtickmarks = 3, title="Left Riemann Sum"): Rplot:=rightbox(x^2, x=1..3, 10, xtickmarks = 3, title="Right Riemann Sum"): Mplot:=middlebox(x^2, x=1..3, 10, xtickmarks = 3, title="Midpoint Riemann Sum"): display(array([Lplot, Rplot])); display(Mplot);

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The graph below gives a more dynamic view of the left-endpoint sums as n increases. Click on the graph, and the toolbar at the top of the window changes. Set the FPS (frames per second) to 1, then press play. with(Student[Calculus1]): ApproximateInt(x^2, 1..3, output=animation, iterations=5, method=left, partition=4, subpartition =all, refinement=halve);

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We can use Maple to calculate the left, right, and midpoint Riemann sums of LUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQtSSRzaW5HRiQ2IywmKiQpSSJ4R0YnIiIjIiIiRjFGMUYxL0YvOyIiIUkjUGlHRiU= and find their errors as shown below. f:= x -> sin(x^2 + 1);

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJiokOSQiIiMiIiJGM0YzRiVGJUYl

The value of the integral is, to eight decimal places, i:= evalf(int(f(x),x=0..Pi), 8);

JCIpMiFbJCopISIp

The left Riemann sum of f(x) when n = 10 is: leftsum(f(x), x=0..Pi, 10); L[10]:=evalf(%,7);

LCQqJkkjUGlHJSpwcm90ZWN0ZWRHIiIiLUkkU3VtR0koX3N5c2xpYkc2IjYkLUkkc2luRzYkRiVGKTYjLCYqJkkiakdGKiIiI0YkRjMjRiYiJCsiRiZGJi9GMjsiIiEiIipGJiNGJiIjNQ==

JCIoUlE8IiEiJw==

The error in this left Riemann sum approximation is E[L]:= i - L[10];

JCEpJCplLkchIik=

The right Riemann sum of f(x) when n = 10 is: rightsum(f(x), x=0..Pi, 10); R[10]:=evalf(%,6);

LCQqJkkjUGlHJSpwcm90ZWN0ZWRHIiIiLUkkU3VtR0koX3N5c2xpYkc2IjYkLUkkc2luRzYkRiVGKTYjLCYqJkkiakdGKiIiI0YkRjMjRiYiJCsiRiZGJi9GMjtGJiIjNUYmI0YmRjg=

JCInOHlmISIn

The error in this right Riemann sum approximation is E[R]:= i - R[10];

JCIpMm5jSCEiKQ==

The midpoint Riemann sum of f(x) when n = 10 is: middlesum(f(x), x=0..Pi, 10); M[10]:=evalf(%,6);

LCQqJkkjUGlHJSpwcm90ZWN0ZWRHIiIiLUkkU3VtR0koX3N5c2xpYkc2IjYkLUkkc2luRzYkRiVGKTYjLCYqJiwmSSJqR0YqRiYjRiYiIiNGJkY1RiRGNSNGJiIkKyJGJkYmL0YzOyIiISIiKkYmI0YmIiM1

JCInangqKSEiJw==

The error in this midpoint Riemann sum approximation is E[M]:= i - M[10];

JCEnJEhHJSEiKQ==

The midpoint case is the most accurate, which is true in general as well. It's clear from the graphs above that the left and right Riemann sums tend to cause over- and under-estimation, while the midpoint rule splits the difference.

Since the left and right endpoint sums tend to err on different sides of the truth, averaging them seems like a good idea: 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