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Determine the area of the curve using six rectangles from $\"\"$ to $\"\"$ .

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Number of intervals $\"\"$ .

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Area is the sum of areas of the $\"\"$ rectangles.

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Width of the each rectangle : $\"\"$ .

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The rectangle repeat for $\"\"$ and $\"\"$ .

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

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Lower estimates :

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Redraw the graph using left endpoints :

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$\"\"$

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Area under the graph using left endpoints :

\

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$\"\"$ .

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Observe the graph:

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The value of the function at different values of $\"\"$ are

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$\"\"$ and $\"\"$ .

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

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$\"\"$

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Area of the graph of the function using left endpoints is $\"\"$ .

\

\

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Upper estimates :

\

Redraw the graph using right endpoints:

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$\"\"$

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

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Area under the graph using right endpoints :

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$\"\"$ .

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$\"\"$ .

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Observe the graph:

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The value of the function at different values of $\"\"$ are

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$\"\"$ and $\"\"$ .

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

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$\"\"$

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Area of the graph of the function using right endpoints is $\"\"$ .

\

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(iii)

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Mid point estimates :

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Redraw the graph using midpoints:

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$\"\"$

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Mid points are $\"\"$ and $\"\"$ .

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Mid point estimation is

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$\"\"$ .

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Observe the graph:

\

The value of the function at different values of $\"\"$ are

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$\"\"$ and $\"\"$ .

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

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$\"\"$

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Area of the graph of the function using mid points is $\"\"$ .

\

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(b)

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The graph of the function is decreasing function.

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From the graph of lower estimate $\"\"$ , it is observed that the bars are above the graph of the curve.

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Therefore, $\"\"$ is an overestimate of the area.

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(c)

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The graph of the function is decreasing function.

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From the graph of left end estimate $\"\"$ , it is observed that the bars are below the graph of the curve.

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Therefore, $\"\"$ is an underestimate of the area.

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(d)

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The function is decreasing function.

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Since the $\"\"$ is an overestimate of the area and $\"\"$ is an overestimate of the area.

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Hence we consider $\"\"$ as the best estimation.

\

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Area under the graph using left endpoints is

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$\"\"$

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Area of the graph of the function using left endpoints is $\"\"$ .

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Area under the graph using right endpoints is

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$\"\"$

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Area of the graph of the function using right endpoints is $\"\"$ .

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Area under the graph using Mid points is

\

$\"\"$

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Area of the graph of the function using mid points is $\"\"$ .

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(b) $\"\"$ is an overestimate of the true area.

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(c) $\"\"$ is an underestimate of the true area.

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(d) $\"\"$ is the best estimation.