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I've added inline derivation comments with source links to all three integrate functions. Here's a summary of the formulas and how to verify them independently.

LinearSpline.integrate

Each segment uses the local-offset linear form:

f_k(x) = C0[k] + C1[k] * (x - xk)

where xk = XValues[k] is the left knot and C0[k] = y(xk) (the y-value at the knot, not a global y-intercept). Integrating by the power rule:

∫[x1,x2] f_k dx = C0[k]*(x2-x1) + C1[k]*((x2-xk)2 - (x1-xk)2) / 2

References: [Power rule]((en.wikipedia.org/redacted) · [FTC]((en.wikipedia.org/redacted)

Verification script (Python/SciPy):

from scipy.interpolate import make_interp_spline
import numpy as np
x = np.array([0., 1., 2., 3., 4.])
y = 2 * x  # y = 2x
spl = make_interp_spline(x, y, k=1)  # linear spline
print(spl.integrate(0, 4))   # expect 16.0
print(spl.integrate(1, 3))   # expect 8.0

Step.integrate

Constant function per segment: f_k(x) = C0[k]. Integral is just the rectangle area:

∫[x1,x2] C0[k] dx = C0[k] * (x2 - x1)

References: [Power rule (n=0 case)]((en.wikipedia.org/redacted)

Verification (Python):

# Step function: y = 3 everywhere on [0,3]
# ∫[0,3] 3 dx = 9

CubicSpline.integrate

The coefficients are stored in global x (not local (x - xk) form):

f_k(x) = a·x3 + b·x2 + c·x + d
  where C0_3[4k] = a, C0_3[4k+1] = b, C0_3[4k+2] = c, C0_3[4k+3] = d

Antiderivative by the power rule:

F_k(x) = a·x4/4 + b·x3/3 + c·x2/2 + d·x
∫[x1,x2] f_k dx = F_k(x2) - F_k(x1)

References: [Power rule]((en.wikipedia.org/redacted) · [FTC]((en.wikipedia.org/redacted)

Verification script (Python/SciPy):

from scipy.interpolate import CubicSpline
import numpy as np
x = np.array([1., 2., 3., 4.])
y = x ** 2  # y = x2
cs = CubicSpline(x, y)
print(cs.integrate(1, 4))   # expect 21.0  (∫[1,4] x2 dx = [x3/3]14 = 64/3 - 1/3 = 21)
print(cs.integrate(1, 2))   # expect 7/3 ≈ 2.333

Note: SciPy's CubicSpline stores coefficients in local (x - xk) form internally — the formula used here is equivalent but in global coordinates, which is how this library's CubicSplineCoef is structured.

Cross-segment accumulation (all three)

When [xVal1, xVal2] spans multiple segments, partial integrals are accumulated using the [additivity of the definite integral]((en.wikipedia.org/redacted)

∫[a,b] f = ∫[a,c] f + ∫[c,b] f

The new comments are now inline in Interpolation.fs alongside each function.

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