$\"\"$

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The rectangular is bounded with $\"\"$ - axis.

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The equation of semi-circle is $\"\"$ .

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Observe the figure,

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Consider length of the rectangular is $\"\"$ .

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The width of the rectangular is $\"\"$ .

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The area of the rectangular is $\"\"$ .

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Substitute $\"\"$ and $\"\"$ values.

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$\"\"$ .

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Find the maximum area.

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$\"\"$ .

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Differentiate on each side with respect to $\"\"$ .

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$\"\"$ .

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$\"\"$

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Product rule of derivatives : $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Find the maximum of $\"\"$ value $\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ in length and width of the rectangular maximum area.

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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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The length of the rectangular is maximum area $\"\"$ .

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The width of the rectangular is maximum area $\"\"$ .

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$\"\"$

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The length of the rectangular is maximum area $\"\"$ .

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The width of the rectangular is maximum area $\"\"$ .