<Text-field style="_cstyle269" layout="Heading 1">Some helpful Maple commands</Text-field>

Lab 6 -- Integration

Objectives:

1. gain expertise in setting up Riemann sums to estimate definite integrals,

2. reinforce the connection between the limit of a Riemann sum and the definite integral,

3. explore using area arguments to calculate definite integrals, and

4. preview the Fundamental Theorem of Calculus.

<Text-field style="_cstyle269" layout="Heading 1">Some helpful Maple commands</Text-field>

Sums

You may choose between two methods to enter 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 and evaluate the sum using Maple...

<Text-field style="Heading 2" layout="Heading 2">typing in Maple commands</Text-field>

The format of the command is 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.

Sum((i/2)^2,i=1..n);

We can get the value of this sum by asking Maple for the value of the last output or by entering the same Sum command with a small s. The command simplify is often useful, particularly when Maple has trouble evaluating the limit of a sum.

value(%);

sum((i/2)^2,i=1..n);

simplify(%);

<Text-field style="Heading 2" italic="true" layout="Heading 2">using the menu options </Text-field>

You'll find the sigma symbol under "Expression" in the left menu.

Put your cursor in an input cell ( [> ) click on the sigma symbol, and complete the relevant pieces.

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

(Try it now.)

Note that Maple attempts to immediately evaluate sums entered this way.

Integrals

Again you have your choice of methods (typing commands or using the menu) for entering definite integrals.

<Text-field style="Heading 2" layout="Heading 2">typing in Maple commands</Text-field>

As with sums, there's a difference between using a captital I (Int) and using a small i (int).

Using Int (capital I) gets you the symbolic representation (this helps to check your typing); using int (lower case i) calculates the integral.

The form of the command is 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

NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNj8tSSNtaUdGJTY5USJmRigvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GKC8lJXNpemVHUSMxMkYoLyUlYm9sZEdRJmZhbHNlRigvJSdpdGFsaWNHUSV0cnVlRigvJSp1bmRlcmxpbmVHRjgvJSpzdWJzY3JpcHRHRjgvJSxzdXBlcnNjcmlwdEdGOC8lK2ZvcmVncm91bmRHUShbMCwwLDBdRigvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYoLyUnb3BhcXVlR0Y4LyUrZXhlY3V0YWJsZUdGOy8lKXJlYWRvbmx5R0Y4LyUpY29tcG9zZWRHRjgvJSpjb252ZXJ0ZWRHRjgvJStpbXNlbGVjdGVkR0Y4LyUscGxhY2Vob2xkZXJHRjgvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYoLyUqbWF0aGNvbG9yR0ZELyUvbWF0aGJhY2tncm91bmRHRkcvJStmb250ZmFtaWx5R0YyLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGKC8lKW1hdGhzaXplR0Y1LUkjbW9HRiU2M1EjOj1GKC8lJWZvcm1HUSZpbmZpeEYoLyUmZmVuY2VHRjgvJSpzZXBhcmF0b3JHRjgvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYoLyUncnNwYWNlR0ZbcC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobWF4c2l6ZUdRKWluZmluaXR5RigvJShtaW5zaXplR1EiMUYoLyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJTBmb250X3N0eWxlX25hbWVHRlgvJSVzaXplR0Y1LyUrZm9yZWdyb3VuZEdGRC8lK2JhY2tncm91bmRHRkctRl9vNjNRIidGKC9GY29RKHBvc3RmaXhGKEZlb0Znby9Gam9RMnZlcnl0aGlubWF0aHNwYWNlRigvRl1wUSQwZW1GKEZecEZgcEZicEZlcEZocEZqcEZccUZecUZgcUZicUZkcUYsRmZxLUZfbzYzUSI7RihGYm9GZW8vRmhvRjsvRmpvRl5yRlxwRl5wRmBwRmJwRmVwRmhwRmpwRlxxRl5xRmBxRmJxRmRxLUZfbzYzUTEmSW52aXNpYmxlVGltZXM7RihGaXFGZW9GZ29GW3JGXXJGXnBGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHFGZHItRl9vNjNRIiNGKEZpcUZlb0Znb0ZbckZdckZecEZgcEZicEZlcEZocEZqcEZccUZecUZgcUZicUZkcS1GLTY5USVUaGlzRihGMEYzRjZGOUY8Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ25GaW5GXG9GZHItRi02OVEoY29tbWFuZEYoRjBGM0Y2RjlGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRmluRlxvRmRyLUYtNjlRJ2NsZWFyc0YoRjBGM0Y2RjlGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRmluRlxvRmRyLUYtNjlRJk1hcGxlRihGMEYzRjZGOUY8Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ25GaW5GXG9GZnEtRi02OVEic0YoRjBGM0Y2RjlGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRmluRlxvLUZfbzYzRmZyRmJvRmVvRmdvRmNyRl1yRl5wRmBwRmJwRmVwRmhwRmpwRlxxRl5xRmBxRmJxRmRxLUYtNjlRJ21lbW9yeUYoRjBGM0Y2RjlGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRmluRlxvRmlzLUYtNjlRI29mRihGMEYzRjZGOUY8Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ25GaW5GXG9GaXMtRi02OVEpd2hhdGV2ZXJGKEYwRjNGNkY5RjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZpbkZcb0Zpc0YsRmlzLUYtNjlRJHdhc0YoRjBGM0Y2RjlGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRmluRlxvLUZfbzYzUSIuRihGYm9GZW9GZ29GY3JGXXJGXnBGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHE3I0MkPkkiZkdGKC5GXXUiIiI=

Int(f(x),x=a..b);

Int(4*x-x^3,x=0..2); value(%);

int(4*x-x^3,x=0..2);

Int(4*t-t^3,t=0..x); value(%);

<Text-field style="Heading 2" layout="Heading 2">using the menu options</Text-field>

The symbol for integrals is also in the "Expression" menu. Note that there are two integral symbols: for definite integrals, use the second.

Put your cursor in an input cell ( [> ) click on the integral symbol, and complete the relevant pieces.

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

Note: be sure the variable (x above, t below) is what you intend it to be.

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

(Try one now.)

<Text-field style="_cstyle271" layout="Heading 1">Lab 6 Homework</Text-field>

The problems for Lab 6 may be found on the course page (Lab 6 Homework, lab06hw_f06.doc ). You will complete, print, and submit the Word document (not any Maple work).

This example is designed to help you with #1 and #2.

I will be using the integral 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.

First, let n = 10. The width of each subinterval is 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(8-2)/10 = 6/10. The endpoints have the form 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. The terms of the sum are 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. For a right-hand sum, we add these terms for 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from 1 to 10; for a left-hand sum we add the terms for 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 from 0 to 9.

f:=x->x+3;

Sum(f(2+6*i/10)*6/10,i=1..10); value(%); evalf(%);

The sum with n = 50:

Sum(f(2+6*i/50)*6/50,i=1..50); value(%); evalf(%);

The sum with n = 100:

Sum(f(2+6*i/100)*6/100,i=1..100); value(%); evalf(%);

The sum with n subintervals, and the limit of this sum:

Sum(f(2+6*i/n)*6/n,i=1..n); value(%); simplify(%); limit(%,n=infinity);

The definite integral:

Int(f(x),x=2..8); value(%);

Comment: keep in mind that your explanations are the key part of these problems.

The function needed for #3:

(Enter the input provided.)

f:=x->piecewise(x<1,1,x>=1 and x < 2, -x+2, x>=2 and x < 4, sqrt(1-(x-3)^2),x>4,-x+4);

'f(x)'=f(x);

Calculate 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.

Here is the relevant graph.

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

This example is designed to help you understand what to do for #4: (Caution: pay careful attention to which variables are used in which places.)

Int(cos(t),t=0..x);

value(%);

diff(%,x);

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C. Wyels, F '03, updated F'06.