Math Visualizations

The Fundamental Theorem of Calculus

The FTC connects differentiation and integration. It states:

Part 1: If $F(t) = \int_a^t f(\tau) \, d\tau$, then $F'(t) = f(t)$.

Part 2: $\int_a^b f(t) \, dt = F(b) - F(a)$, where $F' = f$.

Speed and Distance

In this visualization:

• $v(t) = 3 + 2t$ is the speed function
• $d(t) = 3t + t^2$ is the distance function (integral of speed)

The FTC says:

$$d(t) = \int_0^t v(\tau) \, d\tau \quad \text{and} \quad d'(t) = v(t)$$

• The area under the speed curve from 0 to $t$ equals the distance traveled $d(t)$
• The slope of the distance curve at any point equals the speed at that point

Try This

1. Drag the point on the speed graph to change the time
2. Toggle the area under the speed curve — it equals the distance value
3. Watch the distance graph — the dot shows total distance traveled
4. Notice that the area under speed = distance, and the slope of distance = speed

Real-Life Applications

• Odometers — integrating speed over time gives distance traveled
• Energy consumption — integrating power over time gives total energy
• Population growth — integrating birth rate over time gives total population change