Understanding Velocity-Time Graphs
In a velocity-time graph, the area under the curve holds significant physical meaning, specifically relating to the concept of distance traveled.

Area Under the Graph Represents Distance
- The area under a velocity-time graph quantifies the total distance covered by an object during a specific time interval.
- This is because the velocity (y-axis) multiplied by time (x-axis) gives the distance.

How to Calculate the Area
- The area can be calculated by breaking the graph into geometric shapes (rectangles, triangles, or trapezoids) and summing their areas.
- For example:
- A rectangle represents constant velocity.
- A triangle represents acceleration (change in velocity).

Mathematical Representation
- Mathematically, if you have a graph where velocity \( v \) is plotted against time \( t \), the distance \( d \) can be expressed as:
\[
d = \int v \, dt
\]
- This integral sums up all the infinitesimal areas under the curve, giving the total distance.

Why Other Options Are Incorrect
- **Acceleration (a)**: This is represented by the slope of the velocity-time graph, not the area.
- **Speed**: Although speed is the absolute value of velocity, it does not directly relate to the area under the curve.
- **Time**: Time is simply the duration along the x-axis and does not correspond to the area.
In summary, the area under a velocity-time graph accurately represents the total distance traveled, reinforcing the fundamental relationship between velocity, time, and distance in physics.