Integral Workshop: The Fundamental Theorem

MTH-117 · Semester I · Mathematics I

intermediate25 minAvailable

Limits of quotients

Guided practice — solve each problem, check, then generate a fresh one.

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Formula to apply

lim_{x→a} (x² − a²)/(x − a) = 2a

Problem

Evaluate lim_{x→2} (x² − 4) / (x − 2).

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Working

Lab brief

Evaluate definite integrals with the Fundamental Theorem of Calculus, using antiderivatives of polynomials and trigonometric functions.

Theory & concepts

Fundamental Theorem of Calculus

If F is any antiderivative of f (i.e. F′ = f), then ∫[a to b] f(x) dx = F(b) − F(a).

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Integration as the inverse of differentiation

The reverse power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ −1. A definite integral then evaluates the antiderivative at the two limits and subtracts.

For sine: ∫sin(kx) dx = −cos(kx)/k + C, so ∫[0 to b] sin(kx) dx = (1 − cos(kb))/k.

Don't forget C — but not here

Indefinite integrals carry the constant +C; definite integrals cancel it, which is why F(b) − F(a) works for any choice of antiderivative.

∫[0 to 3] (2x + 5) dx
F(x) = x² + 5x → F(3) − F(0) = 24.
∫[0 to π] sin(2x) dx
(1 − cos(2π))/2 = 0 — the area above and below the axis cancel.
Explain the cancellation
Why does ∫[0 to 2π] sin(x) dx equal zero while the area between the curve and the x-axis is positive?

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