Rectangular Cross-Sections With Variable Height Ratios

From a Base Chord to a Rectangle

A solid can be described by the shapes of its cross-sections. In this tutorial, each cross-section is a rectangle, and one side of the rectangle is a chord of a region in the coordinate plane. The other side, called the rectangle’s height here, is specified as a multiple of that chord. That multiple may be constant or may vary with the slice position.

The key is to find the chord length first. Once the chord is known, the stated height-to-base ratio determines the rectangle’s other side, and the rectangle’s area gives the integrand for the volume.

Finding the Chord and the Ratio

For vertical slices, the cross-section at \(x\) uses the vertical chord in the base region. If the region extends from a lower curve \(y=L(x)\) to an upper curve \(y=U(x)\), its chord length is

For horizontal slices, the chord runs horizontally. If its left and right endpoints are \(x=L(y)\) and \(x=U(y)\), its length is \(b(y)=U(y)-L(y)\). In either orientation, the base is a length in the coordinate plane; the rectangle’s height is the other length specified by the ratio.

Let \(r(x)\) denote the height-to-base ratio at \(x\). The statement “the height is three times the base,” for example, means \(r(x)=3\) and \(h(x)=3b(x)\). If the ratio is given by a function, such as \(r(x)=1+x/4\), use that function at each slice position. A ratio is dimensionless: it compares two lengths measured in the same units.

Why the Volume Integral Has This Form

At each position, the cross-section is a rectangle, so its area is its base times its height. Since the height is \(r(x)b(x)\), the area is \(r(x)b(x)^2\). To obtain volume, imagine dividing the solid into thin slices perpendicular to the \(x\)-axis. A slice of thickness \(\Delta x\) has approximate volume \(A(x)\Delta x\). Adding these contributions and taking the limit gives the definite integral of cross-sectional area.

This formula applies when the slices run from \(x=a\) to \(x=b\), the chord and ratio describe the rectangles throughout that interval, and the resulting area is integrable. For horizontal slices, use \(y\) as the variable and integrate with respect to \(y\). The integral adds areas times thickness, so its units are cubic units.

Worked Examples

A Common Mistake: Using the Ratio as the Height

A ratio tells how the height compares with the base. It is not the height itself. In the square-root example, the ratio is \(1+x/4\), but the chord is \(\sqrt{x}\). The actual rectangle height is \((1+x/4)\sqrt{x}\), so its area is \((1+x/4)(\sqrt{x})^2\).

A wrong calculation would treat \(1+x/4\) as the height and write \(A(x)=(1+x/4)\sqrt{x}\). That would give

but this uses the ratio as a length and omits a factor of the chord. The correction is to calculate \(h(x)=r(x)b(x)\) first, then multiply the base by that height. The correct area is \(r(x)b(x)^2\), giving the volume \(\frac{40}{3}\) cubic units.

A Reliable Setup

Before integrating, check that the geometry has been translated correctly:

• Identify whether slices are vertical or horizontal, and use the corresponding coordinate as the integration variable.
• Find the chord by subtracting the lower endpoint from the upper endpoint, or the left endpoint from the right endpoint.
• Write the height as the stated ratio times the chord.
• Form the rectangular area by multiplying the chord by that height.
• Integrate the cross-sectional area over the interval of slice positions.

In particular, the squared chord in \(A=r b^2\) arises because the chord appears once as the rectangle’s base and again as part of its height. Keeping the base, height, and ratio distinct prevents many setup errors.