Surface Area of z equals xy Inside a Circle

Find the surface area of the part of the function $z = xy$ that lies inside the circle $x^2 + y^2 = 1$ using double integrals.

This problem involves finding the surface area of a part of a surface defined by the function $z = xy$ over a specific domain, namely, the interior of a unit circle defined by $x^2 + y^2 = 1$. The problem is addressed through the application of double integrals, a fundamental concept within the realm of multivariable calculus.

One approach to tackle this problem is to understand the surface in a three-dimensional space and how it is confined by the given circle in the xy-plane. Visualizing this graphically can be useful to comprehend the boundaries and the surface's interaction with these boundaries. The process involves setting up a double integral to account for each infinitesimal piece of the surface, then summing them over the circular region.

The strategy often includes a switch to polar coordinates, which is advantageous given the circular boundary described by $x^2 + y^2 = 1$. In polar coordinates, this transforms into the region 0 to 2 pi for theta and 0 to 1 for r, simplifying the boundaries and the integrand. Through careful evaluation of the double integral, which captures surface area components in terms of the derivatives of the surface function $z$, one can arrive at the solution efficiently. Understanding these concepts is pivotal in solving similar problems involving surface areas over defined regions in space.