jmc

Question

The shaded region shown consists of 11 unit squares and rests along the

x

-axis and the

y

-axis. The shaded region is rotated about the

x

-axis to form a solid. In cubic units, what is the volume of the resulting solid? Express your answer in simplest form in terms of

π

.

Accepted Answer

The resulting solid is the union of two cylinders: one whose radius is 5 units and whose height is 1 unit (the squares shown in light gray produce this cylinder), and the other whose radius is 2 units and whose height is 3 units (shown in dark gray). The sum of these volumes is

π (5)^{2} (1) + π (2)^{2} (3) = 37 π

cubic units.