Express the area of the given surface as an iterated double...

Express the area of the given surface as an iterated double integral in polar coordinates, and then find the surface area.
The portion of the cone $z=\sqrt{x^{2}+y^{2}}$ that lies inside the
cylinder $x^{2}+y^{2}=2 x$

Key Concepts

Determining the Region of Integration

Identifying the correct domain of integration is crucial when setting up a double integral. In this context, the cylindrical boundary, originally given by an equation in Cartesian coordinates, must be converted to polar coordinates. The conversion allows for a clear description of the limits on the radial coordinate r and the angular coordinate ?, which ensures that the integration is performed over the specific region where the two surfaces intersect.

Surface Area Element for a Graph

When computing the surface area of a surface defined as z = f(x, y), one key concept is the surface area element. This involves finding the expression ?(1 + (?z/?x)² + (?z/?y)²) for the infinitesimal piece of the surface. This factor, when multiplied by the differential area element in the domain (dx dy), gives the proper scaling to account for how the surface stretches compared to the xy?plane.

Conversion to Polar Coordinates

Many surfaces exhibit symmetry that is more naturally described in polar coordinates rather than Cartesian coordinates. Converting to polar coordinates involves substituting x = r cos ? and y = r sin ?, with the Jacobian of the transformation contributing a factor of r (i.e., dx dy = r dr d?). This conversion simplifies the integration process over circular or radially symmetric regions.

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