Single Variable Integration

Explore definite integration numerically and analytically. Visualize Riemann sums, midpoint rules, and trapezoids converging towards the exact area under the curve.

Interactive Sandbox

Numerical Methods Integration Visualizer

Select a mathematical function and approximation method. Drag handles directly on the x-axis or adjust sliders to see the numerical shapes, summation elements, and absolute error rates.

f(x) = sin(x) + 1.5

1. Select Function f(x)

2. Select Summation Rule

Subintervals (N) 10
Lower Boundary (a) 1.00
Upper Boundary (b) 3.00
Drag handles 'a' and 'b' on the grid x-axis to alter the integration range. Watch the error drop as N increases.
Algebraic Solver

Integration Steps Solver

Trace the exact analytical integration step-by-step using the Fundamental Theorem of Calculus, and compare it with the active numerical approximation.

Definite Integral Steps


Values Summary


Exact Analytical Area (I)

I = F(b) - F(a) = 0.0000

Numerical Approximation (A_approx)

A = 0.0000

Absolute & Percentage Error

Error = 0.0000 (0.00%)

Evaluation

Integration & Limits Quiz

Test your conceptual understanding of Riemann integration, properties of limits, and basic antiderivatives.

Question 1 of 5 Score: 0/0

Quiz question...

Correct Answer!

Explanation text...

Integration Rules


  • FTC Part 2: ∫[a, b] f(x)dx = F(b) - F(a)
  • Linearity: ∫ f + g dx = ∫ f dx + ∫ g dx
  • Inversion: ∫[a, b] f(x)dx = -∫[b, a] f(x)dx