Volume of Solids: 3D Visualization Tools

Interactive Desmos 3D widgets for visualizing how volumes are computed using the washer method, shell method, and cross-section method. Each widget lets you change the functions, bounds, and axis of revolution to build intuition for how these integrals work.

These widgets are fully interactive. Rotate the 3D view by dragging, zoom with scroll, and adjust sliders in the Desmos expression panel to change functions and bounds. Feel free to make a copy in Desmos and customize the examples for your own class or study session. For a more detailed walkthrough, see the companion blog post.

Washer Method

The washer method computes volume by integrating the area of annular (ring-shaped) cross sections perpendicular to the axis of revolution. The volume of each thin washer is $\pi(R^2 - r^2)\,dx$ (or $dy$), where $R$ is the outer radius and $r$ is the inner radius.

Washers (x-axis)
Revolution about the x-axis. Outer and inner radii defined by two curves as functions of x.
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Washers (Horizontal Axis)
Revolution about a general horizontal axis (not necessarily x = 0). Adjust the axis position with a slider.
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Washers (y-axis)
Revolution about the y-axis. Radii are defined by functions of y (or inverse functions).
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Washers (Vertical Axis)
Revolution about a general vertical axis. Slide the axis of revolution left or right to see how the solid changes.
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Shell Method

The shell method computes volume by integrating the surface area of thin cylindrical shells parallel to the axis of revolution. Each shell has volume $2\pi r \cdot h \cdot \Delta x$ (or $\Delta y$), where $r$ is the distance from the axis and $h$ is the shell's height.

Shells (x-axis)
Cylindrical shells parallel to the x-axis. Shell radius is the vertical distance from the axis.
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Shells (Horizontal Axis)
Shells parallel to a general horizontal axis. Adjust the axis position to see how shells reshape.
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Shells (y-axis)
Cylindrical shells parallel to the y-axis. Shell radius is the horizontal distance from the axis.
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Shells (Vertical Axis)
Shells parallel to a general vertical axis. Move the axis of revolution and watch the shells respond.
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Cross-Section Method

The cross-section method computes volume by integrating the area of slices taken perpendicular to an axis. The shape of each slice (square, semicircle, equilateral triangle, etc.) determines the area formula, and the side length is typically the distance between two curves.

Cross Sections (Squares)
Slices perpendicular to the x-axis with square cross sections. Side length equals the vertical distance between the two curves.
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Cross Sections (Semicircles)
Slices with semicircular cross sections. The diameter of each semicircle spans the gap between the two curves.
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Cross Sections (Equilateral Triangles)
Slices with equilateral triangle cross sections. The base of each triangle spans the distance between the curves.
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