Build section areas and split changing boundaries

Try

Two plane curves can describe a base region for a three-dimensional solid, but they do not specify its volume until the cross-section shape is known. Suppose the base lies between y = x and y = x² for 0 ≤ x ≤ 1. At a fixed x, the vertical segment inside the base has length d = x − x². Predict whether using that segment as a semicircle’s diameter gives the same area as using it as a radius.

Sketch the two curves and one vertical segment. A cross section perpendicular to the x-axis lies in a plane of constant x. Its dimensions are determined by that segment, while its shape extends out of the base plane. Distinguish the one-dimensional slice width d, the two-dimensional cross-section area A(x), and the three-dimensional volume obtained by integrating A over x.

Predict

First suppose every cross section is a rectangle whose height is twice its base width. At x = 1/2 the base segment has length d = 1/4, so the rectangle has dimensions 1/4 by 1/2 and area 1/8. Now suppose the segment is the diameter of a semicircle. Its radius is 1/8, giving area π/128. Predict the ratio of the two cross-section areas before changing x.

Section specification	Dimensions from d	Area
Rectangle, height twice base	Base d, height 2d	2d²
Semicircle, diameter d	Radius d/2	πd²/8
Incorrect semicircle interpretation	Radius treated as d	πd²/2

The last row represents a different solid. It overstates every intended semicircle area by a factor of four; halving a diameter halves the radius and quarters the squared-radius factor.

Experiment

In the live explorer, move the slice through the base region. Predict its dimensions and area before inspecting the result. Switch between rectangles and semicircles, keeping the same base. Verify the midpoint values above, then inspect an endpoint where the two boundary curves meet and the slice degenerates to zero area.

Build a cross-section before integrating

The base lies between y=x² and y=x for 0≤x≤1. At a selected x, the strip has width d=x−x². Perpendicular sections are either rectangles whose height is twice d, or semicircles whose diameter is d. Predict how the area formula changes before revealing calculations.

Section shape Rectangle: height twice the widthSemicircle: width is the diameterSlice location x

The two panels use different zoom levels, each labeled. Within either panel, the horizontal and vertical coordinate units have the same length. In the section, u measures distance across the base strip and z measures height above it. All lengths share one unit; area has square units and volume cubic units. This diagram shows a single perpendicular section, not a solid of revolution.

Text geometry: at x=0.5, the base runs from y=0.25 to y=0.5, so d=0.25. The rectangle has width d and height 2d=0.5.

Reveal area and volumeReset cross-section

Changing either control hides old calculations. The geometry is a supplied exact model; drawing samples only approximate the curved outline. No saved experiment, completion or mastery is awarded. Keep the setup and justification in your written response.

Record the base width, section specification and area at one additional input. Draw the slice yourself, or give a labeled coordinate/dimension description. A fixed base graph does not determine the height of a rectangle unless the height rule is stated. Similarly, a semicircle drawing without a diameter label leaves an important geometric ambiguity.

Understand

For these live examples, d² = (x − x²)² = x² − 2x³ + x⁴. Its integral from 0 to 1 is 1/3 − 1/2 + 1/5 = 1/30. The rectangle solid therefore has volume 2/30 = 1/15 cubic coordinate units. The semicircle solid has volume (π/8)(1/30) = π/240. Since the cross-section areas have constant ratio 16/π at every interior slice, the volumes have the same ratio. This comparison is valid because the two solids use the same x interval and base widths.

If the base coordinates are lengths in centimeters, d is in centimeters, A is in square centimeters, and A dx has cubic-centimeter units. Integrating d alone gives the area of the base region, not either solid’s volume. The slice definition controls the power and coefficient in the integrand.

Now examine a different two-dimensional area problem, separate from the live solid: the region between y = x³ − x and y = 0 on [−1,1]. Solve for every intersection: x(x − 1)(x + 1) = 0 gives −1, 0 and 1. The middle crossing changes which curve is above the other.

Interval	Test input	Sign of x³ − x	Nonnegative vertical width
(−1,0)	−1/2	Positive	x³ − x
(0,1)	1/2	Negative	x − x³

Sketch both lobes and label the upper and lower boundary on each interval. The area is the integral of x³ − x from −1 to 0 plus the integral of x − x³ from 0 to 1. Each contribution is 1/4, for total area 1/2 square coordinate units. Integrating x³ − x over the whole symmetric interval instead gives zero because the signed contributions cancel. That signed integral is correct for its own question, but it is not the area of the two lobes.

Apply

For a fresh pair of curves, solve their intersection equation and check that every root in the requested interval has been found. Split at any point where the upper/lower order changes. On each piece, label a representative slice and form upper minus lower. Only then combine the integrals. A graphing window can suggest crossings, but algebra, a justified sign argument or an explicitly bounded numerical procedure is needed to support completeness.

For a solid built on such a changing-boundary region, first define the nonnegative base width as the distance between the curves. Then apply the section’s geometry. With rectangular height twice the width, the area is twice the square of that width; with semicircle diameter equal to the width, the area is π/8 times its square. Squaring may make the final algebraic expression agree on both pieces, but that algebraic convenience does not remove the need to justify the underlying dimensions and region.

Explain the difference between three plausible answers: an integral of the signed difference, an integral of its absolute value, and an integral of a cross-section area. They respectively describe signed plane accumulation, plane area, and solid volume when their units and hypotheses fit. The same boundary curves can appear in all three calculations without making the resulting quantities interchangeable.

Continue

In practice, submit a sketch or complete slice description beside each setup. Include intersection points, interval order, dimensions, units and a check of at least one representative slice. An exact antiderivative cannot repair an incorrectly defined region or a diameter that was mistaken for a radius.