Final answer:
To express the double integral as a single integral with respect to r, we integrate the density over a spherical shell surface area, simplify to a single variable, and calculate the result using a calculator, ensuring the final answer complies with significant figure rules.
Step-by-step explanation:
To express the double integral in terms of a single integral with respect to r, we need to combine the contributions at each radius by integrating the circular slices or rings. From the provided clues, we understand that charge density varies as a function of radius, and hence the integration across spherical shells is required. The expression given suggests that we should compute the integral of the density ar' across a spherical shell of surface area 4πr'^2 and thickness dr'.
Following the steps provided, a direct answer may not be clear without the full integral presented. However, the process is to substitute the differential arclength dı with r θ dθ and to combine it with the radius-dependent density to obtain a single-variable integral with respect to r. To evaluate the integral, once the appropriate substitutions and simplifications are made, one would input the resultant single-variable integral into a calculator and calculate the numerical result to four decimal places.
It's important to note that the provided information suggests approximation based on significant figures, implying that the final numerical answer should be reported with the appropriate number of significant digits, usually determined by the least precisely known value.