$\"\"$

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The equations are $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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The length of the beam is $\"\"$ .

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The volume of the beam is $\"\"$ .

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

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Substitutte $\"\"$ and $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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Substitute $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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

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

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$\"\"$

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$\"\"$

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

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$\"\"$

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$\"\"$

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Therefore, the volume of the beam is $\"\"$

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$\"\"$

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Find the weight of the beam.

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The density of the concrete is $\"\"$

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The weight of the beam is $\"\"$ .

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Substitute corresponding values.

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$\"\"$

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$\"\"$

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Therefore, the weight of the beam is $\"\"$

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$\"\"$

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(b)

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Find the centroid of the cross section of the beam.

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The centroid of the beam is $\"\"$ .

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The beam is symmetric about $\"\"$ -axis, hence the value of the $\"\"$ .

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The value of $\"\"$ .

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Substitute $\"\"$ .

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$\"\"$ .

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The value of $\"\"$

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$\"\"$ .

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Apply formula : $\"\"$ .

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$\"\"$

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$\"\"$

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

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$\"\"$

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$\"\"$ .

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Therefore, the centroid of the beam is $\"\"$ .

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$\"\"$

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(a) The volume of the beam is $\"\"$ and the weight of the beam is $\"\"$

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(b) The centroid of the beam is $\"\"$ .