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Interactive Integration Visualization

Welcome to the Integration Explorer

This interactive visualization helps you understand integration as the area under a curve. Explore Riemann sums (approximations using rectangles or trapezoids) and see how they approach the exact integral as the number of subdivisions increases.

What you can explore:

Polynomial functions - Learn integration with f(x) = x²
Trigonometric functions - Explore f(x) = sin(x) and its integral
Exponential functions - Discover f(x) = e^x integration
Rational functions - Understand f(x) = 1/x integration
Linear functions - Simple integration with f(x) = x

How to Use This Visualization

Interactive Features:

• Select Function Type - Choose from 5 different functions to explore
• Adjust Bounds - Set the lower bound (a) and upper bound (b) for integration
• Choose Riemann Type - Left, Right, Midpoint, or Trapezoidal approximation
• Change Number of Rectangles - See how more rectangles improve accuracy
• Toggle Visualizations - Show/hide exact area and Riemann approximation

What You'll See:

• Function Curve (colored) - The original function f(x)
• Exact Area (semi-transparent) - The actual area under the curve
• Riemann Rectangles (red/purple) - Approximation using rectangles or trapezoids
• Integration Bounds (red dashed lines) - The limits a and b
• Approximation vs Exact - Compare Riemann sum with exact integral value
• Error Calculation - See how close the approximation is

Function Type

Polynomial function - Power Rule: ∫x²dx = x³/3

Riemann Sum Type

Lower Bound (a): 0.00

Upper Bound (b): 2.00

Number of Rectangles: 10

More rectangles = better approximation

Show Exact AreaShow Riemann Approximation

Integration Results

Riemann Sum Approximation

2.660000

Using 10 rectangles (midpoint method)

Exact Integral Value

2.666667

∫_0.00^2.00 x² dx

Error (Difference)

0.006667

Excellent approximation!

Integration Information

Bounds: [0.00, 2.00]

Width (Δx): 0.2000

Riemann Type: Midpoint

Key Concepts

Definite Integral: The area under the curve between two bounds, represented by ∫_a^b f(x) dx

Riemann Sum: An approximation of the integral using rectangles or trapezoids. As n → ∞, the approximation approaches the exact value.

Methods: Left/Right use endpoints, Midpoint uses center, Trapezoidal uses average of endpoints.

Fundamental Theorem of Calculus

∫_a^b f(x) dx = F(b) - F(a)

Where F(x) is an antiderivative of f(x). The definite integral equals the difference of the antiderivative evaluated at the bounds.

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