Algebra • Topic 30

Integrals

Integration is the reverse of differentiation. While explicit integration is rare in algebra competitions (unlike calculus competitions), integral inequalities are a sophisticated tool.

Fundamental Theorem of Calculus

\int_{a}^{b} f^{'} (x) d x = f (b) - f (a)

Techniques for Bounds

1. Estimating Sums

Integrals can bound sums of monotonic functions.

\int_{1}^{n + 1} f (x) d x \leq k = 1 \sum n f (k) \leq f (1) + \int_{1}^{n} f (x) d x

(assuming

f

is decreasing).

Example (Harmonic Series)

Estimate

H_{n} = 1 + \frac{1}{2} + \dots + \frac{1}{n}

Proof. $ln (n + 1) = \int_{1}^{n + 1} \frac{1}{x} d x < \sum_{k = 1}^{n} \frac{1}{k} < 1 + \int_{1}^{n} \frac{1}{x} d x = 1 + ln n$ . So $H_{n} \approx ln n$ . ∎

2. Integral Inequalities

Cauchy-Schwarz for Integrals:

(\int_{a}^{b} f (x) g (x) d x)^{2} \leq (\int_{a}^{b} f (x)^{2} d x) (\int_{a}^{b} g (x)^{2} d x)

Average Value

The average value of

f

[a, b]

\frac{1}{b - a} \int_{a}^{b} f (x) d x

. MVT for Integrals states

f

assumes its average value at some point

c

Practice Problems

Exercise (Problem 1)

Prove that

\int_{0}^{1} x^{n} (1 - x)^{m} d x = \int_{0}^{1} x^{m} (1 - x)^{n} d x

Exercise (Problem 2)

Estimate

\sum_{k = 1}^{100} \frac{1}{k}

using integrals.

Exercise (Problem 3)

Prove

\int_{0}^{π /2} sin^{10} x d x < \int_{0}^{π /2} sin^{2} x d x