Trapezoidal Rule Calculator
🔷 Trapezoidal Sum Calculator – Approximate Area Under a Curve Easily
The Trapezoidal Sum Calculator helps you approximate the area under a curve by using the trapezoidal rule, a numerical integration method that estimates the definite integral of a function over an interval.
📘 What Is the Trapezoidal Rule?
The Trapezoidal Rule approximates the area under a curve by dividing the region into trapezoids instead of rectangles (like in Riemann sums). ∫abf(x) dx≈Δx2[f(x0)+2f(x1)+2f(x2)+⋯+2f(xn−1)+f(xn)]\int_{a}^{b} f(x) \, dx \approx \frac{\Delta x}{2} \left[f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)\right]∫abf(x)dx≈2Δx[f(x0)+2f(x1)+2f(x2)+⋯+2f(xn−1)+f(xn)]
Where:
- a,ba, ba,b = interval bounds
- nnn = number of subintervals
- Δx=b−an\Delta x = \frac{b – a}{n}Δx=nb−a
- x0,x1,…,xnx_0, x_1, \dots, x_nx0,x1,…,xn are points on the interval
🔢 How the Calculator Works:
- Input a function (e.g.,
f(x) = x^2) - Choose interval limits:
aandb - Enter number of trapezoids:
n - Returns:
- Step-by-step trapezoidal formula
- Numerical result of the approximation
- (Optional) Graph of the trapezoids under the curve
✅ Example
Estimate the area under f(x)=x2f(x) = x^2f(x)=x2 from x=0x = 0x=0 to x=2x = 2x=2 with 4 trapezoids. Δx=2−04=0.5\Delta x = \frac{2 – 0}{4} = 0.5Δx=42−0=0.5 Trapezoidal Sum≈0.52[f(0)+2f(0.5)+2f(1.0)+2f(1.5)+f(2)]=0.25[0+2(0.25)+2(1)+2(2.25)+4]=0.25×10=2.5\text{Trapezoidal Sum} ≈ \frac{0.5}{2}[f(0) + 2f(0.5) + 2f(1.0) + 2f(1.5) + f(2)] = 0.25 [0 + 2(0.25) + 2(1) + 2(2.25) + 4] = 0.25 × 10 = 2.5Trapezoidal Sum≈20.5[f(0)+2f(0.5)+2f(1.0)+2f(1.5)+f(2)]=0.25[0+2(0.25)+2(1)+2(2.25)+4]=0.25×10=2.5
❓ FAQs – Trapezoidal Sum Calculator
🔹 When is the trapezoidal rule more accurate than Riemann sums?
It’s usually more accurate than Left/Right Riemann sums because it accounts for the sloping nature of functions.
🔹 Can the calculator work with both polynomials and trig functions?
Yes — it supports most common mathematical expressions (e.g., sin(x), e^x, ln(x), etc.)
🔹 What if the function is decreasing or nonlinear?
It still works. The trapezoidal rule gives a better approximation than rectangles for curves with slopes.
🔹 How does the number of trapezoids affect accuracy?
Higher n (more subintervals) → smaller trapezoids → more accurate result.
🔹 Is this method exact for linear functions?
Yes — the trapezoidal rule gives exact results for straight-line functions (like f(x) = 2x + 3).