Integration as Area Under Curve
Visualizing Riemann sums and the fundamental theorem of calculus
Explore how integration calculates the area under a curve by adjusting the number of rectangular strips. Observe how increasing the number of strips improves the approximation of the exact area!
Current Calculation:
Riemann sum approximation:
∫ab f(x) dx ≈ Σ f(xi) · Δx
Approximate area: 5.333
Exact area: 5.333
The Concept of Integration as Area
Key Concepts:
Integration measures the area under a curve between two points:
- The Riemann sum approximates the area using rectangles
- As the number of rectangles (n) increases, the approximation improves
- In the limit as n→∞, we get the exact area under the curve
- This is the fundamental concept behind definite integrals
Visual Interpretation:
The green rectangles show the area approximation. As you increase the number of strips:
- The rectangles better fit the curve
- The approximation becomes more accurate
- The sum approaches the exact integral value
