Let region R be the union of two non-overlapping regions R1 and R2 in the xy-plane. Each region is revolved about the x-axis to create solids S1 and S2, respectively, and the entire region R is revolved about the x-axis to create solid S. Which statement best justifies why Vol(S)=Vol(S1)+Vol(S2)?
a) Because, by Pappus’s centroid theorem, the volume of a solid of revolution equals the area of the region times the distance its centroid travels; therefore, adding the areas adds the volumes.
b) Because solid S has the same lateral surface area as the combined lateral surface areas of S1 and S2.
c) Because the definite integral of a sum of functions equals the sum of their definite integrals over the same interval.
d) Because rotating a region about the x-axis is a rigid motion, so the volume is unchanged whether the parts are revolved separately or together.