Let region be the union of two non-overlapping regions and in the -plane

MATHEMATICS · Single correct · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Let region

R

be the union of two non-overlapping regions

R_1

and

R_2

in the

xy

-plane. Each region is revolved about the

x

-axis to create solids

S_1

and

S_2

, respectively, and the entire region

R

is revolved about the

x

-axis to create solid

S

. Which statement best justifies why

\operatorname{Vol}(S)=\operatorname{Vol}(S_1)+\operatorname{Vol}(S_2)

? (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

S_1

and

S_2

. (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.

Related theory

Browse the problem library