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

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

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

Let be the region in the first quadrant bounded by the graph of ​, the horizontal line , and the -axis.

The region is the base of a solid. For each , where , the cross-section of the solid taken perpendicular to the -axis is a rectangle whose height is 4 times the length of its base in region . Write an integral expression that gives the volume of the solid.

Let be the region enclosed by the graph of and the horizontal line , as shown in the figure below.

Region is the base of a solid. For this solid, each cross-section perpendicular to the -axis is an isosceles right triangle with a leg in . 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 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.

Let be the region in the first quadrant enclosed by the graphs of and , as shown in the figure.

The region is the base of a solid. For this solid, at each , the cross-section perpendicular to the -axis has an area given by . Find the volume of the solid.

Another solid has the same base . For this solid, the cross-sections perpendicular to the -axis are squares. Write, but do not evaluate, an integral expression for the volume of the solid.