Additional Bounded Solid Problems
Classically, Calculus III texts present a very limited number of bounded solids for students to perform double or triple integration on. The intersecting surfaces are typically resigned to a class of surfaces called “quadric surfaces”. These surfaces are the 3-dimensional analogue of the conic sections and are very important in their own right.
More exotic surfaces can create interesting solids that are still algebraically accessible for students and offer good practice for various integration techniques (especially the use of polar coordinates and integration by parts). You’ll find such examples below for your mathematical pleasure. Enjoy!
Problem 1: A beautiful glass topped solid is formed by the intersection of the two surfaces defined below. Compute the volume of the bounded region if all dimensions are in feet.
Problem 2: A rubber float is shown below. It was constructed by the intersection of the two surfaces defined below. Compute the volume of the float if all dimensions are in inches.
Problem 3: The surfaces defined below were utilized in creating the flying-saucer shaped object pictured. Compute the volume of the bounded solid if all dimensions are in centimeters.
Problem 4: A glass topped planter is formed by the intersection of the two surfaces defined below. Compute the volume of the bounded region if all dimensions are in feet.
Problem 5: A particular metal cap is shown below. It was formed by the intersection of the two surfaces defined below. Compute the volume of the bounded solid if all dimensions are in centimeters.
Problem 6: A fishing bobber is shown below. It is formed by the intersection of the two surfaces defined below. Compute the volume of the bounded solid if all dimensions are in centimeters.
Problem 7: Consider the two surfaces defined by:
A. Compute the volume of the bounded solid.
B. Compute the mass of the bounded solid if the density is proportional to the square of the distance to the z-axis.
Utilizing Symmetry
Problem 8: A synthetic gem stone was formed by the intersection of the two surfaces defined below. Compute the volume of the gem if all dimensions are in centimeters.
Problem 10: A thorn was formed by the intersection of the two surfaces defined below. Compute the volume of the gem if all dimensions are in centimeters.
Problem 11: An air-filled see-through doggie-house was formed by the functions shown below. Compute the volume of the house if all dimensions are in meters.
Problem 12: An octagonal tent is designed with the functions shown below. Compute the volume of the tent if all dimensions are in meters.