The Trapezoidal Rule

The Trapezoidal Rule approximates the definite integral (area under a curve) by dividing the area into trapezoids:

The Trapezoidal Rule formula is:
$$ \int_a^b f(x)\,dx \approx \frac{b-a}{2n} \left[f(x_0) + 2f(x_1) + 2f(x_2) + \dots + 2f(x_{n-1}) + f(x_n)\right] $$

In this visualization:

• The curve is y = x²
• The interval is [0, 4]
• Blue trapezoids show the approximation
• The exact area is 21.33 square units: $$ \int_0^4 x^2\,dx = \left[\frac{x^3}{3}\right]_0^4 = \frac{64}{3} \approx 21.33 $$

Use the slider to see how increasing the number of trapezoids improves the approximation.