Step 1: \ \

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The function is $\"\"$ on interval $\"\"$ .

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Number of rectangles are $\"\"$ .

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The sum of all inscribed rectangle is lower sum.

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$\"\"$ , where $\"\"$ .

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Width $\"\"$ .

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Step 2: \ \

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Find lower Sum.

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$\"\"$

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$\"\"$ .

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Lower sum is $\"\"$ .

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$\"\"$

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Apply summation formula $\"\"$ .

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Apply summation formula $\"\"$ .

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Apply summation formula $\"\"$ .

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Apply summation formula $\"\"$ .

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In this case $\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$

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The function is $\"\"$ on interval $\"\"$ .

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Number of rectangles are $\"\"$ .

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Find upper Sum.

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The sum of all circumscribed rectangle is upper sum.

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$\"\"$ ,

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where $\"\"$ is the right end point. \ \

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$\"\"$

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$\"\"$ .

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Area of upper sum is $\"\"$

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$\"\"$

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Apply summation formula $\"\"$ .

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Apply summation formula $\"\"$ .Apply summation formula $\"\"$ .

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Apply summation formula $\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$

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The function is $\"\"$ on interval is $\"\"$ .

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Number of rectangles are $\"\"$ .

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Using mid point theorem:

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The area is $\"\"$ .

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Consider $\"\"$ .

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Where $\"\"$ , $\"\"$ and $\"\"$ .

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

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$\"\"$ .

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$\"\"$ .

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

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$\"\"$ .

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$\"\"$ .

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$\"\"$ .

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$\"\"$

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Find $\"\"$ values.

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$\"\"$

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

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Using mid point theorem:

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Area = $\"\"$ .

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$\"\"$

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Substitute $\"\"$ values.

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ sq-units.

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Area of the region is $\"\"$ sq units.

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$\"\"$

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Area of the region is $\"\"$ sq-units.

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The function is $\"\"$ on interval $\"\"$ is decreasing.

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$\"\"$ .

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

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Verification:

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .