6.6 Applying Properties of Definite Integrals

Additivity and Reversal Properties

Definite integrals can be split across intervals: ∫[a,c] f(x)dx = ∫[a,b] f(x)dx + ∫[b,c] f(x)dx, for any b between (or even outside) a and c. Reversing limits negates the integral: ∫[a,b] f(x)dx = -∫[b,a] f(x)dx. Also, ∫[a,a] f(x)dx = 0 since there's no width to the interval.

Linearity: Constants and Sums

Integrals are linear operators: constants factor out, ∫[a,b] k·f(x)dx = k·∫[a,b] f(x)dx, and integrals of sums split apart: ∫[a,b] [f(x)+g(x)]dx = ∫[a,b] f(x)dx + ∫[a,b] g(x)dx. This lets you break complicated integrands into simpler, known pieces to evaluate.

Comparing and Bounding Integrals

If f(x) ≥ g(x) on [a,b], then ∫[a,b] f(x)dx ≥ ∫[a,b] g(x)dx — integrals preserve inequalities. If m ≤ f(x) ≤ M on [a,b], then m(b-a) ≤ ∫[a,b] f(x)dx ≤ M(b-a), bounding the integral using the function's min and max values times interval width.

Key terms

Additivity property of integrals

∫[a,c] f(x)dx = ∫[a,b] f(x)dx + ∫[b,c] f(x)dx, for any b.

Reversal property of integrals

∫[a,b] f(x)dx = -∫[b,a] f(x)dx; swapping limits negates the value.

Integral of a constant multiple

∫[a,b] k·f(x)dx = k·∫[a,b] f(x)dx; constants factor out.

Integral bounding property

If m ≤ f(x) ≤ M on [a,b], then m(b-a) ≤ ∫[a,b] f(x)dx ≤ M(b-a).

Integral comparison property

If f(x) ≥ g(x) on [a,b], then ∫[a,b] f(x)dx ≥ ∫[a,b] g(x)dx.

Try an AP-style question

∫₀⁵ f(x) dx = 8 and ∫₀² f(x) dx = 3. What is ∫₂⁵ f(x) dx?

1. 5
2. −5
3. 8/3
4. 11