Work is fundamentally defined as force times distance. When we push a box with constant force F over distance d, the work done is simply W equals F times d. This straightforward formula works perfectly for constant forces. However, in many real situations, the force varies with position. For example, when stretching a spring, the force increases as we pull further. In such cases, our simple formula breaks down, and we need calculus to find the correct answer.
When force varies with position, we encounter several common examples. Spring force follows Hooke's law, F equals k x, where force increases linearly with displacement. Gravitational force decreases with the square of distance, following an inverse square law. These variable forces create a fundamental problem: we cannot simply multiply force times distance because the force is different at every point along the path. To solve this, we divide the path into small segments where force is approximately constant, then sum up all the small work contributions. This approach naturally leads us to integration.