$\"\"$

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The equation of the cylinder is $\"\"$ and the plane is $\"\"$ .

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The radius of the cylinder is 1 and center at origin.

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The radius of the cylinder varies from 0 to 1.

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If a smooth parametric surface S is given by the $\"\"$ then the surface area of S is $\"\"$ .

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Consider $\"\"$ .

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Evaluate $\"\"$ .

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Apply partial derivative with respect to x.

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$\"\"$

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Evaluate $\"\"$ .

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Apply partial derivative with respect to y.

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$\"\"$

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$\"\"$

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Substitute $\"\"$ and $\"\"$ in the surface area formula.

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$\"\"$

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Convert the area from rectangular to polar coordinates.

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$\"\"$

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Region bounded by the cylinder is $\"\"$ .

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$\"\"$

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Area of the surface is $\"\"$ . $\"\"$

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Area of the surface is $\"\"$ .