6.7 The Fundamental Theorem of Calculus and Definite Integrals

Fundamental Theorem of Calculus (Part 2)

The FTC states that if F is an antiderivative of a continuous function f, then ∫ₐᵇ f(x)dx = F(b) − F(a). This lets you evaluate definite integrals exactly using antiderivatives instead of limits of Riemann sums — connecting integration and differentiation as inverse operations.

Using antiderivatives to evaluate integrals

To compute ∫ₐᵇ f(x)dx: find an antiderivative F(x), then plug in the bounds and subtract: F(b) − F(a). The constant of integration cancels out during subtraction, so you can ignore +C when evaluating definite integrals.

Properties of definite integrals

Key properties: ∫ₐᵃ f(x)dx = 0; ∫ₐᵇ f(x)dx = −∫ᵦᵃ f(x)dx; ∫ₐᵇ f + ∫ᵦᶜ f = ∫ₐᶜ f (additivity over intervals); constants factor out. These properties simplify computing accumulated change and let you split or combine integrals over adjacent intervals.

Key terms

Fundamental Theorem of Calculus (Part 2)

∫ₐᵇ f(x)dx = F(b) − F(a), where F is any antiderivative of continuous f.

Why +C disappears in definite integrals

The constant cancels when subtracting F(a) from F(b), since both include the same +C.

∫ₐᵃ f(x)dx

Equals 0 — integrating over a zero-width interval gives no accumulated change.

Reversing bounds property

∫ₐᵇ f(x)dx = −∫ᵦᵃ f(x)dx — swapping limits negates the integral's value.

Additivity of integrals

∫ₐᵇ f(x)dx + ∫ᵦᶜ f(x)dx = ∫ₐᶜ f(x)dx — integrals over adjacent intervals combine.

Try an AP-style question

∫₁³ (2x + 1) dx = ?

1. 10
2. 12
3. 8
4. 7