Exercises computing volume by cross-sectional area.

The base of a solid region is bounded by the curves , and . The cross sections perpendicular to the -axis are squares. Compute the volume of the region.
  • A typical square cross section has side length and area .
  • Possible numerical values of the -coordinates of points in the base range from a minimum value of up to a maximum of .
  • To compute volume, integrate:
Find the volume of the region in three-dimensional space defined by the inequalities
  • Cross sections perpendicular to the -axis are squarerectangulartriangluar with length in the -direction and width in the -direction.
  • The area of a cross section is .
  • To compute volume, integrate:
A right circular cylinder of radius and height is twisted along its axis so that the disk at height is centered on the axis , which corresponds to one full twist along the axis. Compute the volume of this twisted cylinder.
A certain three-dimensional region has a base in the -plane which is bounded above by the graph and below by . Slices perpendicular to the -axis are equilateral triangles whose base lies in the -plane as well. Compute the volume of the region.

Sample Exam Questions

(2018 Midterm 1) Compute the volume of the region in 3-dimensional space which satisfies the inequalities
none of these
(2019 Midterm 1) The inequality defines an ellipse in the -plane whose area is for any positive values of the constants and . Compute the three dimensional volume of the region defined by (Hints won’t reveal until after you choose a response.)