Calculus I · Unit 3B · lesson
Volumes by Slicing
Learn Volumes by Slicing through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Section overview
Area and volumeWhat this section is building
Learn Volumes by Slicing through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Area adds thin rectangles, slicing adds cross-sectional slabs, washers add annular slabs, and shells add thin cylindrical walls.
Sketch the region and axis, test vertical and horizontal slices, and choose the description that stays single-valued with the fewest interval splits.
Measuring a radius from the wrong curve, subtracting boundaries in the wrong order, or mixing a shell radius with its height.
Learning objectives
Model volume as the integral of cross-sectional area.
Volumes by Slicing
A solid is accumulated cross-sectional area
If we know the area of a cross-section at each position, a thin slab has volume approximately equal to cross-sectional area times thickness. Summing the slabs and taking a limit gives or . This principle covers far more than solids of revolution: squares, triangles, semicircles, and other prescribed shapes all fit the same model.
The essential modeling step is translating a length in the base region into the dimension of the cross-section. State whether that length is a side, diameter, radius, base, or height before writing . Units provide a useful check: area units times length units must produce cubic units. If the integrand does not have units of volume per unit input, the geometry has probably been mistranslated.
If a solid has cross-sectional area perpendicular to the -axis, then
A thin slab has approximate volume . The integral adds the slabs in the limit.
A wedge with triangular cross-sections
A solid has base , and the cross-section at is an equilateral triangle with side . Since triangle area is ,
u3b-slicing-01A solid has square cross-sections of side for . Find its volume.
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Cross-sectional area is .
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A solid assembled from cross-sectional slices. A solid is decomposed into thin slabs; each slab volume is approximately A(x)dx. Show a base interval, one cross-section, and stacked slices. Do not imply slices are literally infinitesimal physical objects.
Every relationship in a solid assembled from cross-sectional slices uses written labels together with distinct line styles, markers, or fill patterns; color is never the only carrier of meaning.
Why it matters: Show a base interval, one cross-section, and stacked slices.
A solid assembled from cross-sectional slices. Show a base interval, one cross-section, and stacked slices.
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Original instruction with traceable references.
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