Volumes with Cross Sections (College Board AP® Calculus AB)

Let and be functions defined by and . The graphs of and , shown in the figure above, intersect at and , where .

The region enclosed by the graphs of and is the base of a solid. Cross sections of the solid taken perpendicular to the -axis are squares. Find the volume of the solid.

Let be the region in the first quadrant bounded by the -axis and the graphs of and , as shown in the figure above.

Region is the base of a solid. For the solid, each cross section perpendicular to the -axis is a rectangle, with its height equal to one third of its base. Write, but do not evaluate, an expression involving one or more integrals that gives the volume of the solid.

Let be the region bounded by the -axis, the line , and the graph of , as shown in the figure above.

Region is the base of a solid. For the solid, each cross section perpendicular to the -axis is a semicircle. Find the volume of the solid.

Let and be functions defined by and . The region is the region enclosed by the graphs of and between and where , as shown in the figure above.

The region is the base of a solid. Cross sections of the solid taken perpendicular to the -axis are equilateral triangles. Write, but do not evaluate, an integral that gives the volume of the solid.