Step 1 :

The equation of the cylinder is and the plane is .

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

The radius of the cylinder varies from 0 to 1.

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

Consider .

Evaluate .

Apply partial derivative with respect to x.

Evaluate .

Apply partial derivative with respect to y.

Step 2 :

Substitute and in the surface area formula.

Convert the area from rectangular to polar coordinates.

Region bounded by the cylinder is .

Area of the surface is .

Solution:

Area of the surface is .