$\"\"$

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The surface of machine part is the region between $\"\"$ and $\"\"$ .

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(a)

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

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

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$\"\"$

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Find $\"\"$ .

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The parabola is tangent to the graph of $\"\"$ .

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Equate $\"\"$ .

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$\"\"$

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Therefore $\"\"$ .

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Find $\"\"$ .

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

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$\"\"$

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Therefore, $\"\"$ .

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Therefore the parabola equation is $\"\"$ .

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$\"\"$

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(b)

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Find the area.

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Find the points of intersections.

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Equate the two equations.

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$\"\"$

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$\"\"$ .

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Find the roots using quadratic roots formula.

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$\"\"$

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Substitute $\"\"$ , $\"\"$ and $\"\"$ in the above formula.

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$\"\"$

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

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The area of the surface of the machine part is $\"\"$ .

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Substitute $\"\"$ and $\"\"$ in the above equation.

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$\"\"$

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$\"\"$

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The area of the surface of the machine part is $\"\"$ sq-units.

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$\"\"$

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(a) The value of $\"\"$ is $\"\"$ .

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(b) The area of the surface of the machine part is $\"\"$ sq-units.