Definite integrals, areas and volumes

Definite integrals, areas and volumes · Diagrams · Complete reading

From the source curve to left and right rectangle sums

Compare left and right rectangle sums

Illustrative curve f(x) = 2 + x − x² on [0, 1.5]. Change the strip count and endpoint rule; the source drawing supplies a rising-then-falling shape, not this formula.

Teaching and source notes

The original shows the area A, then left heights f(xᵢ) and right heights f(xᵢ₊₁), with equal width Δx = (b − a)/n. The explorer keeps those choices and adds a midpoint comparison using the existing workbench.

The chosen numerical example rises and then falls. Its exact integral is [2x + x²/2 − x³/3] from 0 to 1.5, which is 3. The formula, bounds and value are illustrative, not extra givens in the source.

On a nonmonotone curve, the endpoint name alone does not prove that a sum is below or above the integral. Inspect the whole interval. Increasing n numerically illustrates convergence; it does not prove the general theorem.

Three source figures reconstructed: area A under a continuous nonnegative curve from a to b, left rectangles of height f(x_i), and right rectangles of height f(x_(i+1)). The curve rises then falls, so endpoint choice alone gives no global under/over estimate. All widths are delta x=(b−a)/n.

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Signed integrals and areas

Equal areas, opposite signed integrals

Use the source sine intervals: [−π, 0], [0, π] and [−π, π]. Compare the signed integral with the total geometric area. Decimal axis labels approximate the exact preset bounds.

Teaching and source notes

The source sine regions each have geometric area 2. Their signed integrals are −2 and +2, so the combined signed integral is 0 while the geometric area is 4. The explorer retains those exact preset intervals in radians.

For a region above the axis, area is the integral. Below the axis, area is the negative of the integral. Across a sign change, split the interval and add the nonnegative areas; taking the absolute value of the combined integral can lose area.

Shaded running areas are numerical estimates, compared with the antiderivative −cos x. Rounded numerical agreement does not replace the analytic evaluation or the sign split.

Sine has equal geometric areas 2 below and above the axis; its signed integrals cancel. General above-axis, below-axis and mixed regions require positive, negative and split area formulas respectively.

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Horizontal strips and the y-axis

A horizontal strip has width g(y) and height delta y. Areas right of the y-axis use a positive integral; areas left use its negative; mixed regions split at the crossing and add positive areas.

Derive upper minus lower

Build area from upper minus lower

Illustrative curves f(x) = 2x − x² and g(x) = x² on [0, 1]. Inspect a strip's two heights before multiplying their difference by its width.

Teaching and source notes

The source derives A = F − G by subtracting the area under the lower curve from that under the upper curve, between their meeting points a and b. The numerical formulas here are labelled illustrative rather than attributed to the original sketch.

For f(x) = 2x − x² and g(x) = x², f − g = 2x(1 − x) is nonnegative on [0, 1]. The curves meet at both endpoints. Integrating the upper curve gives 2/3, the lower gives 1/3, and their difference is 1/3.

The same bounds and consistent upper/lower ordering are essential. If curves cross within an interval, split before summing geometric areas. A finite strip sum is still an approximation.

Between two intersections, upper curve f minus lower curve g gives the enclosed area. The region is shown as the area F under f minus G under g.

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Choose vertical or horizontal strips

Vertical strips use upper f(x) minus lower g(x) between x bounds a,b; horizontal strips use right f(y) minus left g(y) between y bounds c,d. Split if the ordering changes.

A strip rotates into a disc

For rotation about the x-axis, a vertical strip of height y and thickness delta x becomes a disc of radius absolute y. For rotation about the y-axis, a horizontal strip of width x and thickness delta y becomes a disc of radius absolute x.

Recognise cylinders and full cones

Rectangles rotated about either axis produce cylinders with volume pi r squared h. The two triangular regions drawn with an apex on the axis produce full cones with volume one third pi r squared h; a nonzero radius at both ends would instead give a frustum.

Strips rotate into washers about either axis

For ordered nonnegative radii, a vertical strip rotates about the x-axis into a washer of thickness delta x; a horizontal strip rotates about the y-axis into a washer of thickness delta y. Each area is pi times outer-radius squared minus inner-radius squared.

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