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The fundamental theorem of calculus evaluates a definite integral as F(b) − F(a), where F is any antiderivative. Simpson's rule is exact for cubics, so it confirms the result here.
Fundamental theorem
∫ₐᵇ f(x) dx = F(b) − F(a)
F(x) = 0.75x⁴ − x² + x, giving F(3) − F(1) = 54.75 − 0.75 = 54.
Yes. It is a signed area, so regions below the axis count negatively.