Area Bounded by Curve Calculator

Calculator Inputs

Example Data Table

Formula Used

For two curves, the bounded area over [a, b] is:

A = ∫[a,b] |f(x) - g(x)| dx

For signed area, the calculator uses:

S = ∫[a,b] (f(x) - g(x)) dx

Simpson rule uses:

∫[a,b] y dx ≈ (h / 3) [y0 + yn + 4(y1 + y3 + ...) + 2(y2 + y4 + ...)]

Trapezoidal rule uses:

∫[a,b] y dx ≈ h [(y0 + yn) / 2 + y1 + y2 + ... + y(n-1)]

Midpoint rule uses:

∫[a,b] y dx ≈ h Σ y(xi + h / 2)

How to Use This Calculator

1. Enter the upper function as f(x).
2. Enter the lower function as g(x). Use 0 for the x-axis.
3. Set the lower and upper x-limits.
4. Choose Simpson, trapezoidal, or midpoint integration.
5. Choose total bounded area or signed integral.
6. Use auto split when curves cross inside the interval.
7. Press Calculate Area to view results above the form.
8. Use CSV or PDF buttons to save your report.

Supported functions include sin, cos, tan, sqrt, abs, exp, log, log10, pow, min, and max.

Area Between Curves in Calculus

Core Idea

An area bounded by a curve is a closed region on a graph. It can sit between one curve and the x-axis. It can also sit between two different curves. This calculator is designed for both cases. You enter an upper function, a lower function, and an x-range. The tool estimates the area with numerical integration.

Why Bounded Area Matters

Bounded area appears in many calculus problems. It helps measure accumulated change. It also supports physics, economics, probability, and engineering work. A curve can represent speed, cost, density, height, or growth. The area then describes a total amount. For example, area under velocity gives displacement. Area between revenue and cost can estimate profit.

Numerical Method Overview

Many functions do not have easy antiderivatives. Some functions are only known from data. Numerical integration solves this issue. The calculator divides the interval into small parts. It then estimates thin strips and adds them together. Smaller widths usually improve accuracy. Simpson’s method is often accurate for smooth curves. The trapezoidal rule is simple and stable. The midpoint rule can reduce some endpoint error.

Using Upper and Lower Curves

For two curves, the basic expression is f(x) minus g(x). A signed result keeps direction. An absolute result measures the total shaded size. This is important when curves cross. If the difference changes sign, signed areas may cancel. The auto split option searches for crossing points. It then separates the interval into smaller regions. This gives a clearer bounded-area report.

Reading the Graph

The graph shows both functions and sample points. It also helps verify the interval. Use it to check whether the region is closed. If the chosen limits do not enclose a meaningful region, revise them. The table gives sample values, differences, and strip estimates. Export options help save results for homework, reports, and teaching notes.

Best Practice Tips

Always start with a rough sketch. Compare several methods when accuracy matters. Increase the number of subintervals slowly. Watch for vertical asymptotes, jumps, and sharp corners. These features can distort estimates. Use the formula notes to explain each result clearly in your final solution with good units.