Final answer:
The area scaling factor in integrals accounts for how areas change under a variable transformation, often involving a Jacobian determinant in multi-dimensional transformations.
Step-by-step explanation:
When a change of variable is made in an integral, the area scaling factor refers to the factor by which areas transform under the change of variable. Typically, this involves a Jacobian determinant in multi-dimensional cases, which represents the factor by which volume or area scales when transforming coordinates.
In the context provided, the area scaling factor is required to adjust the integral accordingly. The process of changing variables often necessitates the use of substitution which may include a scalar factor that adjusts the dimensions of the integral to retain its physical meaning or value. As the question highlights, integrals can be interpreted as sums of infinitesimals representing the areas of strips, and the change in variables must account for this accordingly.
To calculate the integral effectively and find the area scaling factor, one would need to apply the proper transformation rules and integration techniques that involve the given variables such as r or the change in temperature (ΔT) alongside constants like the coefficient of linear expansion (a).