Surface Area Above a Rectangle for a Function in 3D

Find the surface area of $Z = \sqrt{4 - x^2}$ which is above the region $R$, a rectangle defined by $x \in [0, 1]$ and $y \in [0, 4]$.

This problem requires finding the surface area of a three-dimensional object defined by a function $z = f(x, y)$. Specifically, we are interested in the surface area of the portion of $z = \sqrt{4 - x^2}$ that lies above a specific rectangle in the xy-plane. The rectangle is defined by $x$ values ranging from 0 to 1 and $y$ values from 0 to 4.

To approach this problem, one would typically make use of concepts from surface area calculations in multivariable calculus. This involves setting up a double integral over the given region $R$. The integrand in this situation is derived from the formula for surface area in terms of partial derivatives. Specifically, you would need the square root of 1 plus the squares of the partial derivative with respect to $x$ and the partial derivative with respect to $y$.

Additionally, while working through this type of problem, understanding the geometry and shape of the surface is key. It helps to visualize or sketch the function $z = \sqrt{4 - x^2}$ in the context of its intersection with the $yz$-plane to recognize that it forms a segment of a cylinder with its axis along the $y$-direction. Recognizing this underlying geometric shape can simplify the task of setting up and evaluating the necessary integrals to find the solution.