$\"\"$

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(a)

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The integral of the function is $\"\"$ .

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Number of sub intervals are $\"\"$ .

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Riemann sum is $\"\"$ , Where width of the interval is $\"\"$ and $\"\"$ .

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Here $\"\"$ , $\"\"$ and $\"\"$ .

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Width of the interval is $\"\"$ and $\"\"$ .

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End points of eight subintervals are $\"\"$ and $\"\"$ .

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Right end points are $\"\"$ and $\"\"$ .

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Substitute corresponding values in the formula for Riemann sum.

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$\"\"$

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$\"\"$ .

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$\"\"$

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Determine the right Riemann sums using the graphing utility.

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Graph :

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Graph the function $\"\"$ and rectangles for $\"\"$ at an interval of $\"\"$ :

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$\"\"$

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$\"\"$

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Theorem 4:

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If $\"\"$ is integrable on $\"\"$ , then $\"\"$ , where width of the interval $\"\"$ and $\"\"$ .

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Here $\"\"$ , $\"\"$ and $\"\"$ .

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Width of the interval is $\"\"$ .

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$\"\"$ .

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Substitute $\"\"$ and $\"\"$ in $\"\"$ .

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$\"\"$

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The sum of squares of $\"\"$ natural numbers is $\"\"$ .

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The sum of $\"\"$ natural numbers is $\"\"$ .

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$\"\"$

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$\"\"$

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Graph :

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Graph the Riemann sum under the given intervals as a difference of areas $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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(a) $\"\"$ .

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(b)

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$\"\"$

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(c) Integral result by using theorem 4,

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$\"\"$ .

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(d)

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$\"\"$ .