Mathematics for Physics I

Definition and Interpretation

Integrals represent accumulation and are fundamental to understanding how quantities add up over intervals in physics.

Definite Integral

The definite integral of a function f(x) from a to b is defined as:

\[ \int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x \]

Where:

\(\Delta x = \frac{b-a}{n}\) is the width of each subinterval
\(x_i^*\) is a point in the i-th subinterval

This represents the net signed area between the curve y = f(x) and the x-axis from x = a to x = b.

Indefinite Integral

The indefinite integral of a function f(x) is a family of functions F(x) such that:

\[ \int f(x) \, dx = F(x) + C \]

Where:

F(x) is an antiderivative of f(x), meaning F'(x) = f(x)
C is an arbitrary constant of integration

The indefinite integral represents the general antiderivative of the function.

Interactive visualization of the definite integral as the area under a curve

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus establishes the connection between differentiation and integration, showing that they are inverse operations.

Part 1: Derivative of an Integral

If f is a continuous function on [a, b] and we define:

\[ F(x) = \int_a^x f(t) \, dt \]

Then F is differentiable on (a, b) and:

\[ F'(x) = f(x) \]

This means the derivative of the integral of a function gives back the original function.

Part 2: Evaluation of Definite Integrals

If f is a continuous function on [a, b] and F is any antiderivative of f, then:

\[ \int_a^b f(x) \, dx = F(b) - F(a) \]

This is often written using the notation:

\[ \int_a^b f(x) \, dx = \left[ F(x) \right]_a^b = F(b) - F(a) \]

This provides a practical method for evaluating definite integrals without using the limit of Riemann sums directly.

Example: Evaluating a Definite Integral

Evaluate \(\int_0^2 x^2 \, dx\):

\begin{align} \int_0^2 x^2 \, dx &= \left[ \frac{x^3}{3} \right]_0^2 \\ &= \frac{2^3}{3} - \frac{0^3}{3} \\ &= \frac{8}{3} - 0 \\ &= \frac{8}{3} \end{align}

This represents the area under the curve y = x² from x = 0 to x = 2.

Integration Techniques

Basic Formulas

\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1) \] \[ \int \frac{1}{x} \, dx = \ln|x| + C \] \[ \int e^x \, dx = e^x + C \] \[ \int a^x \, dx = \frac{a^x}{\ln a} + C \]

Trigonometric Integrals

Integration by Substitution

If u = g(x) and du = g'(x)dx, then:

\[ \int f(g(x))g'(x) \, dx = \int f(u) \, du \]

Example:

\begin{align} \int x\cos(x^2) \, dx &= \int \cos(u) \cdot \frac{du}{2} \quad \text{(where } u = x^2 \text{)} \\ &= \frac{1}{2} \int \cos(u) \, du \\ &= \frac{1}{2} \sin(u) + C \\ &= \frac{1}{2} \sin(x^2) + C \end{align}

Integration by Parts

The formula is:

\[ \int u \, dv = uv - \int v \, du \]

Example:

\begin{align} \int x\sin x \, dx &= -x\cos x + \int \cos x \, dx \quad \text{(where } u = x, \, dv = \sin x \, dx \text{)} \\ &= -x\cos x + \sin x + C \end{align}

Partial Fractions

For rational functions \(\frac{P(x)}{Q(x)}\) where degree of P < degree of Q, decompose into simpler fractions.

Example:

\begin{align} \int \frac{1}{x^2-1} \, dx &= \int \frac{1}{(x-1)(x+1)} \, dx \\ &= \int \left( \frac{A}{x-1} + \frac{B}{x+1} \right) \, dx \\ &= \int \left( \frac{1/2}{x-1} - \frac{1/2}{x+1} \right) \, dx \\ &= \frac{1}{2} \ln|x-1| - \frac{1}{2} \ln|x+1| + C \\ &= \frac{1}{2} \ln\left|\frac{x-1}{x+1}\right| + C \end{align}

Trigonometric Substitution

Used for integrals involving \(\sqrt{a^2-x^2}\), \(\sqrt{a^2+x^2}\), or \(\sqrt{x^2-a^2}\).

\begin{align} \sqrt{a^2-x^2} &: \text{ substitute } x = a\sin\theta \\ \sqrt{a^2+x^2} &: \text{ substitute } x = a\tan\theta \\ \sqrt{x^2-a^2} &: \text{ substitute } x = a\sec\theta \end{align}

Improper Integrals

Improper integrals involve either infinite limits of integration or integrands with vertical asymptotes within the interval.

Type 1: Infinite Limits

\[ \int_a^{\infty} f(x) \, dx = \lim_{t \to \infty} \int_a^t f(x) \, dx \] \[ \int_{-\infty}^b f(x) \, dx = \lim_{t \to -\infty} \int_t^b f(x) \, dx \] \[ \int_{-\infty}^{\infty} f(x) \, dx = \int_{-\infty}^c f(x) \, dx + \int_c^{\infty} f(x) \, dx \]

Example:

\begin{align} \int_1^{\infty} \frac{1}{x^2} \, dx &= \lim_{t \to \infty} \int_1^t \frac{1}{x^2} \, dx \\ &= \lim_{t \to \infty} \left[ -\frac{1}{x} \right]_1^t \\ &= \lim_{t \to \infty} \left( -\frac{1}{t} + 1 \right) \\ &= 0 + 1 = 1 \end{align}

Type 2: Discontinuous Integrands

\[ \int_a^b f(x) \, dx = \lim_{\epsilon \to 0^+} \int_a^{c-\epsilon} f(x) \, dx + \lim_{\epsilon \to 0^+} \int_{c+\epsilon}^b f(x) \, dx \]

Where f has a vertical asymptote at x = c, with a < c < b.

Example:

\begin{align} \int_0^1 \frac{1}{\sqrt{x}} \, dx &= \lim_{\epsilon \to 0^+} \int_{\epsilon}^1 \frac{1}{\sqrt{x}} \, dx \\ &= \lim_{\epsilon \to 0^+} \left[ 2\sqrt{x} \right]_{\epsilon}^1 \\ &= \lim_{\epsilon \to 0^+} (2 - 2\sqrt{\epsilon}) \\ &= 2 \end{align}

Convergence and Divergence

An improper integral is said to converge if its limit exists as a finite number. Otherwise, it diverges.

Common tests for convergence include:

p-test: \(\int_1^{\infty} \frac{1}{x^p} \, dx\) converges if p > 1 and diverges if p ≤ 1
Comparison test: If 0 ≤ f(x) ≤ g(x) for x ≥ a, then:
- If \(\int_a^{\infty} g(x) \, dx\) converges, then \(\int_a^{\infty} f(x) \, dx\) also converges
- If \(\int_a^{\infty} f(x) \, dx\) diverges, then \(\int_a^{\infty} g(x) \, dx\) also diverges

Applications in Physics

Area and Volume

The area between two curves f(x) and g(x) from a to b is:

\[ A = \int_a^b |f(x) - g(x)| \, dx \]

The volume of a solid of revolution around the x-axis is:

\[ V = \pi \int_a^b [f(x)]^2 \, dx \]

These concepts are essential for calculating physical quantities like cross-sectional areas and volumes of irregular objects.

Work and Energy

Work done by a variable force F(x) over a displacement from a to b is:

\[ W = \int_a^b F(x) \, dx \]

For a conservative force, this equals the negative change in potential energy:

\[ W = -(U(b) - U(a)) \]

This relationship is fundamental to understanding energy conservation in physical systems.

Electric and Magnetic Fields

The electric potential due to a continuous charge distribution is:

\[ V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \int \frac{\rho(\vec{r'})}{|\vec{r} - \vec{r'}|} \, dV' \]

The magnetic field due to a current-carrying wire is given by the Biot-Savart law:

\[ \vec{B}(\vec{r}) = \frac{\mu_0}{4\pi} \int \frac{I \, d\vec{l} \times (\vec{r} - \vec{r'})}{|\vec{r} - \vec{r'}|^3} \]

These integrals are essential in electromagnetism for calculating fields from distributed sources.

Average Values

The average value of a function f(x) over an interval [a, b] is:

\[ f_{avg} = \frac{1}{b-a} \int_a^b f(x) \, dx \]

This concept is used to calculate average quantities like temperature, pressure, or density over a region.

For periodic functions with period T, the average over one period is:

\[ f_{avg} = \frac{1}{T} \int_0^T f(t) \, dt \]

This is particularly important for analyzing oscillatory phenomena in physics.

Example: Work Done Against Gravity

Calculate the work done in lifting a 10 kg object from the Earth's surface to a height of 1000 m, accounting for the variation of gravitational force with height.

The gravitational force at height h above the Earth's surface is:

\[ F(h) = \frac{GMm}{(R_E + h)^2} \]

Where:

G = 6.67 × 10⁻¹¹ N·m²/kg² (gravitational constant)
M = 5.97 × 10²⁴ kg (Earth's mass)
m = 10 kg (object's mass)
R_E = 6.37 × 10⁶ m (Earth's radius)

The work done is:

\begin{align} W &= \int_0^{1000} F(h) \, dh \\ &= \int_0^{1000} \frac{GMm}{(R_E + h)^2} \, dh \\ &= GMm \int_0^{1000} \frac{1}{(R_E + h)^2} \, dh \\ &= GMm \left[ -\frac{1}{R_E + h} \right]_0^{1000} \\ &= GMm \left( -\frac{1}{R_E + 1000} + \frac{1}{R_E} \right) \\ &= GMm \left( \frac{R_E - (R_E + 1000)}{R_E(R_E + 1000)} \right) \\ &= GMm \left( \frac{-1000}{R_E(R_E + 1000)} \right) \\ &\approx 9.8 \times 10 \times 1000 \times \frac{R_E}{R_E + 1000} \\ &\approx 98000 \times 0.9998 \\ &\approx 97980 \text{ J} \end{align}

This is slightly less than the work calculated using the constant gravity approximation (98000 J), reflecting the decrease in gravitational force with height.

Key Insight:

Integrals provide a powerful tool for calculating cumulative effects in physics, allowing us to handle variable forces, non-uniform distributions, and continuous changes. They enable us to move beyond the simplified constant-value approximations and model the continuous nature of physical phenomena more accurately. Whether calculating work, energy, fields, or flux, integration is the mathematical language for summing infinitesimal contributions across space, time, or other variables.

Practice Problems

Test your understanding of integrals with these practice problems:

Evaluate \(\int x^2\sqrt{x+1} \, dx\).
Find the area between the curves y = x² and y = x³ from x = 0 to x = 1.
Calculate the work done by a force F(x) = 3x² + 2x when moving an object from x = 1 to x = 3.
Determine whether the improper integral \(\int_1^{\infty} \frac{1}{x\ln x} \, dx\) converges or diverges.
A particle moves along a straight line with velocity v(t) = t² - 4t + 3. If the particle is at position s = 2 when t = 0, find its position at t = 3.