AB Calculus - Hardtke
Assignment 6.4: Volume by Cross-Section
Name ______________________________
Due Date: Monday, 2/17
Rather than Volumes of Revolution, now we will look at finding Volumes by Cross-Sections.
Example 1:
 Imagine that the floor of a building is the region bounded by x = y2 and the x = 9.
 A tent or building is formed by placing vertical cross-sections along this base where each cross-section is a
square of perfect size to cross the parabolic region on the floor.
 Each square cross-section is perpendicular to the x-axis, so the thickness of the cross-section is represented by
dx. Find the volume.
*To draw the 3-D figure, it is helpful to drop the x & y axis to allow for a vertical perspective of the z-axis.
Example 2: For the same base as Example 1, each cross-section is a semi-circle. Find the volume.
Example 3: For the same base as Example 1, each cross-section is an equilateral triangle. Find the volume.
Example 4: The base of a solid, S, is an elliptical region with boundary curve 9x2 + 4y2 = 36. Cross-sections
perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base. Find the volume.
2
1. Find the volume if the base of a volume is the region bounded by x = 1 and x = 2 – y and cross-sections are squares
perpendicular to the x-axis. Sketch a graph including one typical cross-section.
2. The base of a certain solid is the triangular region with vertices (0, 0), (1, 1), and (2, 0). Cross-sections perpendicular
to the x-axis are semi-circles. Find the volume of the solid.
Over 
3. On the region bounded by
√ and y = x, a volume by cross-sections is formed by squares perpendicular to the xaxis. Sketch a graph including one typical cross-section and then find the volume of the solid.
4. A solid has a circular base of radius 1. Parallel cross-sections perpendicular to the base are equilateral triangles.
Find the volume of the solid.
AB Calculus - Hardtke
Assignment 6.4: Volume by Cross-Section
SOLUTION KEY
Rather than Volumes of Revolution, now we will look at finding Volumes by Cross-Sections.
Example 1:
 Imagine that the floor of a building is the region bounded by x = y2 and the x = 9.
 A tent or building is formed by placing vertical cross-sections along this base where each cross-section is a
square of perfect size to cross the parabolic region on the floor.
 Each square cross-section is perpendicular to the x-axis, so the thickness of the cross-section is represented by
dx. Find the volume.
*To draw the 3-D figure, it is helpful to drop the x & y axis to allow for a vertical perspective of the z-axis.
Example 2: For the same base as Example 1, each cross-section is a semi-circle. Find the volume.
Example 3: For the same base as Example 1, each cross-section is an equilateral triangle. Find the volume.
Example 4: The base of a solid, S, is an elliptical region with boundary curve 9x2 + 4y2 = 36. Cross-sections
perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base. Find the volume.
AB Calculus - Hardtke
Assignment 6.4: Volumes by Cross-Section
SOLUTIONS
2
1.
Find the volume if the base of a volume is the region bounded by x = 1 and x = 2 – y and cross-sections are squares
perpendicular to the x-axis. Sketch a graph including one typical cross-section.
2.
The base of a certain solid is the triangular region with vertices (0, 0), (1, 1), and (2, 0). Cross-sections perpendicular to the x-axis
are semicircles. Find the volume of the solid.
Over 
3.
On the region bounded by y  x and y = x, a volume by cross-sections is formed by squares perpendicular to the x-axis.
Sketch a graph including one typical cross-section and then find the volume of the solid.
4.
A solid has a circular base of radius 1. Parallel cross-sections perpendicular to the base are equilateral triangles.
Find the volume of the solid.