Calculate the bounded area
Supported functions: sin, cos, tan, asin, acos, atan, sqrt, abs, log, ln, exp, floor, and ceil. Use ^ for powers.
Formula used
Geometric area: A = ∫ab |f(x) − g(x)| dx
Signed integral: I = ∫ab [f(x) − g(x)] dx
When one curve stays above the other, use A = ∫ab [top(x) − bottom(x)] dx. When curves cross, absolute-area mode keeps every strip positive. The calculator estimates these integrals with Simpson's rule or the trapezoidal rule.
How to use this calculator
- Enter the two curve expressions using
xas the variable. - Set the left and right x bounds for the intended region.
- Choose absolute area for physical space between the curves.
- Select a numerical method, slice count, precision, and optional unit label.
- Press Calculate area, then review the result, intersections, chart, and table.
- Download the CSV or PDF after checking the plotted region.
Example data table
Example curves: f(x) = 4 − x2 and g(x) = x2 − 2 over [−2, 2].
| x | f(x) | g(x) | Vertical gap |
|---|---|---|---|
| −2 | 0 | 2 | 2 |
| −1 | 3 | −1 | 4 |
| 0 | 4 | −2 | 6 |
| 1 | 3 | −1 | 4 |
| 2 | 0 | 2 | 2 |
Understand bounded curve regions
The area between two curves measures the space trapped by their graphs. A calculator estimates that space across a selected horizontal interval. You enter two expressions and two x limits. The page evaluates both expressions at many nearby points. It then adds narrow vertical strips to estimate the region. This approach works for lines, polynomials, trigonometric functions, exponentials, and many mixed expressions.
Choose useful limits
Before calculating, inspect the interval carefully. The useful region may begin or end at an intersection. Curves can also cross inside the limits. When they cross, the upper curve changes. Absolute-area mode handles these changes automatically. It measures every strip as a positive amount. Signed mode keeps the direction of subtraction. This helps when you want the net integral instead of geometric area.
Read the numerical estimate
The standard vertical-strip formula is integral from a to b of top function minus bottom function, with respect to x. For crossing curves, use the absolute value of their difference. A numerical method replaces the integral with many small estimates. Simpson's rule usually offers strong accuracy for smooth curves. The trapezoidal rule is simple and useful for comparison. Raising the slice count can improve detail, although difficult functions may still need careful bounds.
Check intersections and graphs
Intersections matter because they describe where the vertical gap becomes zero. The calculator scans the interval and refines detected crossings with bisection. The reported roots are approximate. Check a graph whenever a curve has sharp turns, discontinuities, or several nearby crossings. A graph also shows whether your bounds include the intended region. It can reveal a missing segment before you export results.
Use exports responsibly
Use the sample table to inspect values at plotted x positions. Each row shows both function values and the vertical gap. Download the CSV for spreadsheets or records. Download the PDF for a compact summary. Keep the expressions readable. Write x^2 for powers and sin(x) for trigonometric input. Use parentheses around grouped terms. For example, write (x+1)^2 rather than x+1^2.
Know the limits
Results are numerical approximations, not symbolic proofs. Increase slices when values change noticeably. Avoid intervals across vertical asymptotes. Split a problem into safe intervals when needed. Confirm units before reporting a final answer. If x uses meters, the area uses square meters. Good inputs and sensible bounds produce dependable estimates. Proceed carefully.
Frequently asked questions
What does this calculator find?
It estimates the region between two y-values across a chosen x interval. Absolute mode reports geometric area. Signed mode reports the net value of f(x) minus g(x).
Which expression format should I use?
Use x as the variable. Write powers with ^, such as x^2. Use explicit multiplication, such as 2*x. Group terms with parentheses when needed.
Can the curves cross each other?
Yes. Choose absolute geometric area to keep every strip positive. The calculator also scans for crossing points and lists approximate intersections inside your selected bounds.
Why is the signed result negative?
A negative signed result means the second curve is generally above the first curve across more of the interval. Use absolute mode when you need physical area.
Which numerical method is better?
Simpson's rule is usually more accurate for smooth curves with the same slice count. The trapezoidal rule is easier to compare and can be useful for rough estimates.
How many slices should I choose?
Start with 400 to 600 slices. Increase the count when the result changes noticeably or when curves bend sharply. More slices take slightly longer to calculate.
What do the listed intersections mean?
They are approximate x-values where both functions have equal y-values. They often mark boundaries where the curve above changes or where a region begins or ends.
Can I use trigonometric functions?
Yes. Use sin(x), cos(x), tan(x), and related supported functions. Inputs use radians. Include parentheses around the function argument.
Why did the calculator show a domain error?
The expression may be undefined at a bound or inside the interval. Common causes include division by zero, square roots of negative values, and logarithms of nonpositive values.
How are units handled?
Enter an optional length label such as m or ft. The primary area result is shown in squared units. Leave the field blank for generic square units.
What do CSV and PDF downloads include?
CSV contains the calculation summary and sampled table values. PDF contains a compact result summary, method details, intersection values, and core inputs for convenient sharing.