1. 6.4 Volumes by Cross Sections
Rita Korsunsky

2. Volumes by cross sections:
Let S be a solid bounded by planes that are perpendicular to the x-axis at a and b. If, for every x in [a, b], the cross-sectional area of S is given by A(x), where A is continuous on [a,b], then the volume of S is x y
Calculus Cross Section Link 1
Calculus Cross Section Link 2

3. Example 1: Find the volume of a right pyramid with a square base of side a and altitude h.

4. Example 2: Find the volume of the solid whose base is bounded by the graphs of y = x + 1 and y = x2 - 1, with square cross sections taken perpendicular to the x-axis.
Draw set of axis. Put letters x and y on (+) sides of axis
Since cross-section is perpendicular to x-axis, draw the line through the region, which is parallel to y-axis x y
Draw vertical heights of your cross-
section 1 -1
Connect them with the line segment parallel to the brown line segment 1 -1

5. Example 2: Find the volume of the solid whose base is bounded by the graphs of y = x + 1 and y = x2 - 1, with square cross sections taken perpendicular to the x-axis.
Square’s side length =
y of upper curve – y of lower curve
Square’s area = x y 1 -1

6. y -1 1 x Draw set of axis. Put letters x and y on (+) sides of axis
Example of drawing. Draw the solid whose base is bounded by the graphs of y = x + 1 and y = x2 - 1, with square cross sections taken perpendicular to the x-axis.
Equilateral triangle
Draw set of axis. Put letters x and y on (+) sides of axis
Since cross-section is perpendicular to x-axis, draw the line through the region, which is parallel to y-axis x y 1 -1
Draw vertical heights of your cross-
section
Construct the triangle

7. y -1 1 x Draw set of axis. Put letters x and y on (+) sides of axis
Example of drawing. Draw the solid whose base is bounded by the graphs of y = x + 1 and y = x2 - 1, with square cross sections taken perpendicular to the x-axis.
rectangular
y-axis
Draw set of axis. Put letters x and y on (+) sides of axis
Since cross-section is perpendicular to y-axis, draw the line through the region, which is parallel to x-axis x y 1 -1
Draw vertical heights of your cross-
section
Connect them with the line segment parallel to the brown line segment

8. More examples of how to make Cross-Sectional Diagrams in 3D