Visual Guide: - The x-axis represents Time (t) and the y-axis represents Velocity (v). - The red dashed lines a and b mark the start and end of our journey. - We slice the time window into $N = 12$ equal intervals, each with a duration of $\Delta t = 0.6667\text{s}$. Why this matters: In order to find the area under a curved shape, we must first break it down into manageable sections.

Visual Guide: - Each purple block is a rectangle representing a time slice. Its height is the car's speed at that moment, and its width is $\Delta t$. - The area of one block ($\text{speed} \times \text{time}$) equals the distance covered in that slice. - Adding all rectangle areas together gives our total approximated distance of 88.237 meters. Without Calculus: - Without integration, we can only compute distance if speed is completely constant (like driving exactly 60 km/h the whole time). - If speed changes constantly (accelerating, braking), simple multiplication fails. We would have to guess the distance, which leads to large errors.

Visual Guide: - The emerald green curve traces the car's continuous speedometer profile. - The definite integral represents the *exact* area under this curve. The Convergence Principle: - By increasing the number of partitions ($N$), the blocks become narrower and fit the curve more closely. - Taking the limit as $N \to \infty$ merges the block approximation with the exact definite integral ($88.267\text{ meters}$), bringing the error margin down to $0\%$.