Integration & Area Under Curves

Understand integration as an accumulation of area. The Power Rule states that $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$). See how discrete Riemann sums converge to exact integrals.

Interactive Sandbox

Riemann Sum Convergence Graph

Drag the vertical boundaries **a** and **b** directly on the graph or adjust the sliders. Change interval density $N$ to watch the rectangles fill the curve, matching the exact analytical area.

Curve: y = x²

Power Rule Integration

∫ xn dx = xn+1/(n+1) + C
Intervals Count (N) 8
Lower Boundary (a) 0.50
Upper Boundary (b) 2.50
Drag boundaries 'a' or 'b' on the grid. Observe the percentage error drop as interval counts (N) increase.
Algebraic Solver

Convergence & Error Steps

Compare numerical approximations (Left-endpoint Riemann sums) vs exact analytical integration values.

Analytical Integration Steps


Area Calculations Summary


Exact Area ∫ab f(x) dx

Exact = 5.17

Riemann Sum Approximation (N)

Riemann = 4.82

Approximation Percentage Error

Error = 6.8%

Evaluation

Integration Quiz

Test your knowledge of the power rule, integration bounds, and Riemann sums.

Question 1 of 5 Score: 0/0

Evaluate the indefinite integral ∫ x³ dx.

Correct Answer!

Explanation text...

Integration Formulas


  • ∫ x^n dx = [x^(n+1)] / (n+1)
  • ∫ 1/x dx = ln|x| (Special Case n = -1)
  • Definite Integral: F(b) - F(a)