The area under the curve is important when we look at speed and how quickly something changes speed in math called calculus. Specifically, on a speed-time graph, the area under the curve shows us the total distance an object travels over a certain period.

Key Ideas:

1. Speed Function:

   • If we call the speed of an object $v(t)$, then the distance $d$ it travels from time $a$ to time $b$ can be found using this formula: $d = \int_a^b v(t) \, dt$

2. Acceleration:

   • The acceleration, or how quickly the speed is changing, is shown by the function $a(t)$, which gives us the derivative of the speed. So, we have $a(t) = \frac{dv}{dt}$.
   • The area under the acceleration-time graph helps us understand how speed changes over time.

3. Total Change in Speed:

   • To find out how much the speed changes, denoted as $Δv$, between the two times $t_1$ and $t_2$, we can integrate the acceleration: $Δv = \int_{t_1}^{t_2} a(t) \, dt$

Example:

• Imagine a particle that speeds up at $a(t) = 3t^2$ from time $t=1$ to time $t=3$. The change in speed can be calculated as: $Δv = \int_1^3 3t^2 \, dt = [t^3]_1^3 = 27 - 1 = 26 \text{ m/s}$

This shows how the area under curves in calculus is directly linked to understanding motion in real life.