Understanding Definite Integrals and Riemann Sums

This page introduces the concept of definite integrals and their relationship to Riemann sums. It provides a comprehensive overview of the mathematical foundations and practical applications of these crucial calculus concepts.

Definition: A definite integral of a bounded function f: [a,b] → R is defined as the limit of Riemann sums as the partition of the interval becomes infinitely fine.

The definite integral is formally expressed as:

∫[a to b] f(x) dx = lim[n→∞] Σ[i=1 to n] f(ξᵢ)(xᵢ - xᵢ₋₁)

Where P is a partition of the interval [a,b], and ξᵢ are sample points in each subinterval.

Highlight: The definite integral represents the area under the curve of f(x) bounded by the lines x=a, x=b, y=0, and the graph of f(x).

The page also introduces the concept of Suma Riemanna (Riemann sum), which is a finite approximation of the definite integral. The Riemann sum is given by:

S(P) = Σ[i=1 to n] f(ξᵢ)(xᵢ - xᵢ₋₁)

Vocabulary: The diameter of a partition, denoted as δ(P), is the maximum width of any subinterval in the partition.

The fundamental theorem of calculus is presented, stating that if f(x) is continuous, then it is integrable, and:

∫[a to b] f(x) dx = F(b) - F(a)

Where F(x) is any antiderivative of f(x).

Example: The definite integral ∫[a to b] c dx = c(b-a), where c is a constant.

The page concludes by listing several important properties of definite integrals, including additivity with respect to the interval and linearity.

Integration Techniques and Examples

This page delves into advanced integration techniques, focusing on integration by substitution, and provides several worked examples to illustrate these methods.

Definition: Integration by substitution is a method used to evaluate definite integrals by changing the variable of integration.

The general formula for integration by substitution is:

∫[a to b] f(g(x))g'(x) dx = ∫[g(a) to g(b)] f(u) du

Where u = g(x) is the substitution.

The page presents three detailed examples of definite integral calculations:

1. ∫[0 to π] x cos(x) dx
2. ∫[0 to 2] x√(4-x²) dx
3. ∫[0 to π/2] cos³(x) dx

Example: For ∫[0 to π] x cos(x) dx, the solution uses integration by parts: Result: [x sin(x)]₀ᵖⁱ - ∫[0 to π] sin(x) dx = π sin(π) - (-cos(π) + cos(0)) = 0 + 2 = 2

Highlight: The second example, ∫[0 to 2] x√(4-x²) dx, demonstrates the power of substitution in simplifying complex integrals.

The page concludes with a note on the practical applications of definite integrals, emphasizing their use in calculating areas and volumes in various fields of science and engineering.

Vocabulary: Całka Riemanna wzór (Riemann integral formula) is the mathematical expression that defines the definite integral as the limit of Riemann sums.

This comprehensive guide provides students with a solid foundation in understanding and applying definite integrals, answering questions like "Co mierzy całka?" (What does an integral measure?) and "Do czego służy rachunek całkowy?" (What is the purpose of integral calculus?).