Engineering Mathematics II

Integral Calculus

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Module IV: Integral Calculus. This chapter covers definite and improper integrals, convergence tests, reduction formulae, Beta and Gamma functions, differentiation under the integral sign, and multiple integrals with Jacobians and applications. These tools are extremely important in engineering mathematics and appear frequently in university examinations.

1. Fundamental Theorem of Integral Calculus

The Fundamental Theorem establishes the connection between derivatives and definite integrals:

  • If F(x) is an antiderivative of f(x), then∫ₐᵇ f(x) dx = F(b) - F(a).
  • If F(x) = ∫ₐˣ f(t) dt, thendF/dx = f(x).
  • The theorem allows evaluation of definite integrals using primitive functions.

Diagram: fundamental-theorem-integral.png

2. Mean Value Theorems for Integrals

If f is continuous on [a, b], then there exists c in (a, b) such that:

∫ₐᵇ f(x) dx = f(c) (b - a)

This states the area under the curve equals the area of a rectangle of width (b - a) and height f(c).

Diagram: integral-mean-value.png

3. Improper Integrals and Convergence Tests

An improper integral involves infinite limits or infinite discontinuities.

  • Type I: Infinite limits∫ₐ^∞ f(x) dx = lim (b→∞) ∫ₐᵇ f(x) dx.
  • Type II: Discontinuity at a point.∫ₐᶜ f(x) dx = lim (t→c) ∫ₐᵗ f(x) dx.

Tests of Convergence

  • Comparison Test: compare with known integrable functions.
  • Limit Comparison Test: use limits to compare growth rates.
  • p-integral: ∫₁^∞ 1/x^p dx converges if p greater than 1.

Diagram: improper-integral-types.png

4. Reduction Formulae

Reduction formulae express integrals of higher powers in terms of lower powers. They greatly simplify repeated integration in exams.

  • ∫ (sin^n x) dx reduces to(n-1)/n ∫ (sin^(n-2) x) dx
  • ∫ (cos^n x) dx follows the same pattern.
  • I_n = ∫ (x^n e^x dx) reduces using integration by parts:
    I_n = x^n e^x - n I_(n-1)

Diagram: reduction-formula.png

5. Beta and Gamma Functions

Beta and Gamma functions appear widely in definite integrals, probability distributions and Fourier analysis.

Gamma Function

  • Γ(n) = ∫₀^∞ x^(n-1) e^(-x) dx
  • Γ(n+1) = n Γ(n)
  • Γ(1) = 1, Γ(1/2) = √π

Beta Function

  • B(m, n) = ∫₀¹ x^(m-1) (1-x)^(n-1) dx
  • Relation with Gamma:B(m, n) = Γ(m) Γ(n) / Γ(m + n)

Diagram: beta-gamma-plot.png

6. Differentiation Under the Integral Sign

Also known as Leibniz Rule. Helpful in solving integrals that are otherwise difficult.

d/dα ∫ₐᵇ f(x, α) dx = ∫ₐᵇ ∂f/∂α dx

Example and diagram are easily asked in exams.

Diagram: leibniz-rule-visual.png

7. Double and Triple Integrals

These represent area, volume and mass distributions. Their evaluation often involves change of variables and Jacobians.

Double Integrals

∬_R f(x, y) dAevaluated as iterated integrals.

Triple Integrals

Used for computing volumes and mass distributions:∭_V f(x, y, z) dV.

Diagram: double-integral-region.png

Jacobian for Change of Variables

∬ f(x, y) dx dy = ∬ f(x(u, v), y(u, v)) |J| du dv

|J| is the absolute value of the Jacobian determinant.

Diagram: jacobian-transformation.png

Complex Variables