Answer
See explanation
Work Step by Step
Step 1: In this problem, a single integral property and a double integral property are mentioned. We have to explain how the double integral property generalizes the given single integral property. Step 2: Let us start by stating the given single integral property: If $a, b, c$ are three points, then irrespective of their relative order, we can write the following: \[ \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx \] As long as the function is integrable on the two intervals on the right-hand side. Now, we will state the double integral property that has been referred: If $R$ can be subdivided into regions $R_1$ and $R_2$, then we can write \[ \iint_R f(x, y) \, dA = \iint_{R_1} f(x, y) \, dA + \iint_{R_2} f(x, y) \, dA \] Step 3: The double integral property is the generalization of the single integral property because: If $y$ was a function of $x$, then the double integral property can be simplified into the single integral property. This means that the single integral property represents a special case in which $R, R_1, R_2$ are curves (instead of the general case of areas).
