Calculus I · Unit 3B · lesson

Volumes by Slicing

Concept

Learning objectives

Model volume as the integral of cross-sectional area.

Volumes by Slicing

Explanation

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 $V=\int A(x)\,dx$ or $V=\int A(y)\,dy$ . 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 $A(x)$ . 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 $A(x)$ perpendicular to the $x$ -axis, then

V=\int_a^bA(x)\,dx.

A thin slab has approximate volume $A(x_i^*)\Delta x$ . The integral adds the slabs in the limit.

Application

A wedge with triangular cross-sections

A solid has base $0\le x\le4$ , and the cross-section at $x$ is an equilateral triangle with side $s(x)=\sqrt{x}$ . Since triangle area is $\sqrt3s^2/4$ ,

A(x)=\frac{\sqrt3}{4}x, \qquad V=\int_0^4\frac{\sqrt3}{4}x\,dx=2\sqrt3.

Interactive checku3b-slicing-01

A solid has square cross-sections of side $x$ for $0\le x\le2$ . Find its volume.

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Cross-sectional area is $x^2$ .

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**Visual study**
A solid assembled from cross-sectional slices. Show a base interval, one cross-section, and stacked slices.

After the explanation

Use the section idea

Reading lens

Let the axis and slice orientation determine every distance; top-minus-bottom, right-minus-left, radii, and shell height must all come from the same picture.

Mental model

Area adds thin rectangles, slicing adds cross-sectional slabs, washers add annular slabs, and shells add thin cylindrical walls.

Decision

Sketch the region and axis, test vertical and horizontal slices, and choose the description that stays single-valued with the fewest interval splits.

Common trap

Measuring a radius from the wrong curve, subtracting boundaries in the wrong order, or mixing a shell radius with its height.

Check yourself

Do the slice dimensions remain nonnegative on the full interval, and do their units multiply to area or volume?

Source & rights

Original instruction with traceable references.

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