Position, Velocity, and Work: Integral Calculus in Physics

Summary of Previous Lectures

Introduction to Differential Calculus in Physics

Differential calculus is essential in physics for understanding how physical quantities change over time. In particular, the velocity of an object can be found from the time derivative of its position, and acceleration from the time derivative of velocity, especially in one-dimensional motion.

• Position ($x(t)$): The location of an object as a function of time.

• Velocity ($v(t)$): The rate of change of position with respect to time, $v(t) = \frac{d}{dt}x(t)$.

• Acceleration ($a(t)$): The rate of change of velocity with respect to time, $a(t) = \frac{d}{dt}v(t) = \frac{d^2}{dt^2}x(t)$.

Integral Calculus

2.1 Why Integrals in Physics?

Integral calculus is deeply intertwined with physics because it allows us to determine quantities such as position from velocity and velocity from acceleration. Integrals are also used to calculate work, energy, and other physical quantities that involve summing infinitesimal contributions over a region or interval.

• Physical Application: Determining position from velocity, and acceleration from velocity, using integrals.

• Conceptual Understanding: Integrals provide a way to sum up continuous changes, which is essential for understanding physical processes.

2.2 What is an Integral?

An integral represents the summation of quantities over a given interval. The basic idea is to add up infinitesimal slices under a curve to find the total area, which can represent physical quantities like distance, work, or energy.

• Riemann Sums: Approximating the area under a curve by summing the areas of rectangles.

• Definite Integral: The exact area under a curve as the width of rectangles approaches zero.

Example: Summing numbers 1 to 5: $S = \sum_{i=1}^5 n_i = 1 + 2 + 3 + 4 + 5 = 15$

Area under a curve using Riemann sums:

• $A = \sum_{i=1}^n f_i \Delta x$

• As $\Delta x \to 0$, $A = \int_a^b f(x)dx$

2.3 Indefinite Integrals

An indefinite integral is the antiderivative of a function, representing a family of functions differing by a constant. It is written as:

• $\int f(x)dx = F(x) + C$

• Antiderivative: If $f(x) = 2x$, then $F(x) = x^2$.

• General Form: $F(x) + C$ includes all possible antiderivatives.

Examples of Indefinite Integrals:

• Polynomial: $\int 3x^2 dx = x^3 + C_1$

• Trigonometric: $\int 2\cos(2x) dx = \sin(2x) + C_2$

• Exponential: $\int e^x dx = e^x + C_3$

2.4 Definite Integrals

Definite integrals compute the exact (numeric) value of the area under a curve between two points, using the fundamental theorem of calculus:

• $\int_a^b f(x)dx = F(b) - F(a)$

• No arbitrary constant in definite integrals; only the difference between endpoints matters.

Example: For $f(x) = x$ from $a = 0$ to $b = 1$:

• Antiderivative: $F(x) = \frac{1}{2}x^2$

• Definite integral: $\int_0^1 x dx = \frac{1}{2}(1^2) - \frac{1}{2}(0^2) = \frac{1}{2}$

Position and Velocity

3.1 Determining Velocity from Acceleration in 1D

Integral calculus allows us to determine velocity from a known acceleration function in one dimension. The indefinite integral of acceleration gives velocity, plus a constant of integration.

• $v(t) = \int a_x(t)dt = A_x(t) + C_v$

• Definite Integral for Change in Velocity: $\Delta v_x = \int_a^b a_x(t)dt = A_x(b) - A_x(a)$

• Final Velocity: $v_x = v_{x0} + \int_{t_0}^t a_x(t)dt$

3.2 Determining Position from Velocity in 1D

Similarly, integrating the velocity function yields the position function. The indefinite integral of velocity gives position, plus a constant of integration.

• $x(t) = \int v_x(t)dt = V_x(t) + C_x$

• Definite Integral for Displacement: $\Delta x = \int_a^b v_x(t)dt = V_x(b) - V_x(a)$

• Final Position: $x = x_0 + \int_{t_0}^t v_x(t)dt$

Displacement vs. Total Distance:

• Displacement: $\Delta x = x - x_0$

• Total Distance: $|\Delta x| = \int_{t_0}^t |v_x(t)|dt$

Important Distinction: Displacement is the net change in position, while total distance is the sum of all movement regardless of direction.

Table: Types of Integrals in Physics

Type	Definition	Application
Indefinite Integral	$\int f(x)dx = F(x) + C$	Find antiderivative; general solution
Definite Integral	$\int_a^b f(x)dx = F(b) - F(a)$	Compute exact value over interval
Riemann Sum	$A = \sum_{i=1}^n f_i \Delta x$	Approximate area under curve

Key Formulas

• Velocity from Position: $v(t) = \frac{d}{dt}x(t)$

• Acceleration from Velocity: $a(t) = \frac{d}{dt}v(t)$

• Position from Velocity: $x(t) = \int v_x(t)dt$

• Velocity from Acceleration: $v(t) = \int a_x(t)dt$

• Definite Integral: $\int_a^b f(x)dx = F(b) - F(a)$

Summary

Integral calculus is a foundational tool in physics, enabling the calculation of position, velocity, and other physical quantities from their rates of change. Understanding both definite and indefinite integrals, as well as their applications to kinematics, is essential for solving problems in introductory physics.