Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13
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The scalar line integral of x over the right unit semicircle equals two

Example

Let γ(t)=(cos⁡t,sin⁡t) for −π/2≤t≤π/2, oriented from the bottom to the top of the right unit semicircle. For the scalar field f(x,y)=x,

∫γx ds=2.

Facts & Assumptions

Given: The path and scalar field in the Example.

[L1]

A scalar line integral is ∫f(γ(t))∥γ′(t)∥2dt on a C1 piece (Scalar line integrals with respect to arc length and vector-field line integrals).

[L2]

Sine and cosine have derivatives cos⁡t and −sin⁡t, satisfy sin⁡2t+cos⁡2t=1, and have values sin⁡(π/2)=1 and sin⁡(−π/2)=−1 (The derivatives of sine and cosine are cosine and minus sine, Parity and the Pythagorean identity for sine and cosine, Quarter-turn values and shifts by pi/2 and pi).

[L3]

For a continuous function whose interior derivative admits an integrable extension, Newton-Leibniz integrates that extension to the endpoint increment (Newton–Leibniz needs only continuity on [a,b], differentiability on (a,b), and a Riemann-integrable extension of the interior derivative).

[L4]

Scalar line integrals are unchanged by orientation reversal (Scalar line integrals are parametrization-independent; vector line integrals retain orientation and change sign when it reverses).

Verification

technique · direct
1.1

By [L2], γ′(t)=(−sin⁡t,cos⁡t) and ∥γ′(t)∥2=1, while f(γ(t))=cos⁡t.

givenL2algebra
2.1

By [L1], [L2], and [L3], ∫γx ds=∫−π/2π/2cos⁡t dt=sin⁡(π/2)−sin⁡(−π/2)=2.

step 1.1L1L2L3algebra
3.1

Reversing the path leaves the value 2 unchanged by [L4], confirming that the scalar ds integral does not depend on orientation.

step 2.1L4∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 103 results over 24 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources