The Fundamental Theorem of Calculus bridges the gap between differentiation (slopes) and integration (areas).
1. Differentiation: The red dashed line shows the derivative at x = 4. The slope is 0.25, representing the instantaneous rate of change of the function y = √x.
2. Integration: The green shaded region represents the definite integral from 2 to 4. The area is approximately 3.45.
To apply this to the graph, we view the curve \( y = \sqrt{x} \) as a rate of change, \( f'(x) \). The shaded area represents the total accumulated change in the original function \( f(x) \).
\[ \text{Area} = \int_{2}^{4} \underbrace{\sqrt{x}}_{f'(x)} \, dx = \underbrace{\left[ \frac{2}{3}x^{3/2} \right]_{2}^{4}}_{f(b) - f(a)} \approx 3.45 \]The theorem tells us that the accumulated area under the derivative curve (green region) is exactly equal to the net change in the original function's value.