Area Between Curves Calculator
Compute the area under a single curve, the area between two curves, or the area enclosed by a polar curve — using numerical integration (Simpson’s Rule), so it works with any function you can type, not just ones with a simple antiderivative. Enter a function using x as the variable (e.g. x^2, sin(x), sqrt(x)+1) and the interval you want.
This calculator finds the area associated with a function’s graph — the kind of problem that comes up in AP Calculus AB/BC and college Calculus I/II: the area under a single curve over an interval, the area enclosed between two curves, and the area swept out by a polar curve between two angles. It works by numerical integration (Simpson’s Rule) rather than symbolic calculus, so it can handle any function you can type — polynomials, trig, exponential, logarithmic, or combinations of them — even when the function doesn’t have a simple antiderivative.
Because this tool integrates numerically, it reports a decimal number rather than an exact symbolic antiderivative or a step-by-step “integrate, then evaluate” solution. For the area-between-curves mode, you enter the interval yourself; it does not automatically search for where the two curves intersect. If you need the exact symbolic antiderivative and full algebraic steps, a computer-algebra tool is the right choice — this calculator is built for getting an accurate numeric answer quickly and for checking your own by-hand work.
How to Enter a Function
Type the expression using x as the variable. Supported operators are +, -, *, /, and ^ for exponents, with parentheses for grouping. Supported functions are sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, sqrt, abs, exp, ln (natural log), log (base 10), log2, cbrt, floor, ceil, and round, plus the constants pi and e. You can write 2x or 3sin(x) without a multiplication sign — the calculator inserts it automatically.
Area Under a Curve
Formula
Area = ∫ab f(x) dx
Where a and b are the interval’s lower and upper bounds and f(x) is the function.
Net Area vs. Total Area
If f(x) dips below the x-axis anywhere in [a, b], the plain definite integral (net area) counts that part as negative, so it can partly or fully cancel out the area above the axis. Total area treats every part as positive by integrating |f(x)| instead, which matches what most “area under the curve” problems are actually asking for. This calculator reports both.
Worked Example
Find the area under f(x) = x² from x = 0 to x = 2:
Area = ∫02 x² dx = [x³/3] from 0 to 2 = 8/3 ≈ 2.667
Since x² never goes negative on [0, 2], the net and total areas are the same here.
Area Between Two Curves
Formula
Area = ∫ab |f(x) − g(x)| dx
Using the absolute value means the formula gives the correct enclosed area whether f(x) stays above g(x) for the whole interval or the two curves cross one or more times inside it.
Worked Example
Find the area between f(x) = x² and g(x) = √x from x = 0 to x = 1 (the two curves meet at x = 0 and x = 1, with √x above x² in between):
Area = ∫01 (√x − x²) dx = [2x3/2/3 − x³/3] from 0 to 1 = 2/3 − 1/3 = 1/3 ≈ 0.333
Finding the Bounds Yourself
This calculator does not solve for intersection points automatically. Set f(x) = g(x) and solve, or graph both functions, to find where they cross, then use those x-values as a and b.
Polar Area
Formula
Area = ½ ∫αβ r(θ)² dθ
Where r(θ) is the polar function and α, β are the starting and ending angles. Enter the angles in either degrees or radians — the calculator converts degrees to radians before integrating.
Worked Example
Find the area enclosed by the full cardioid r(θ) = 1 + cos(θ) from θ = 0 to θ = 2π:
Area = ½ ∫02π (1 + cosθ)² dθ = 3π/2 ≈ 4.712
What This Calculator Doesn’t Cover
This is a numeric tool, not a computer-algebra system: it does not produce exact fractions, symbolic antiderivatives, or a step-by-step “integrate term by term” solution, and for area-between-curves it needs you to supply the integration bounds rather than finding intersection points on its own. It also assumes the function is defined (no division by zero, no square root of a negative number, and so on) across the interval you enter — if it isn’t, the calculator reports an error instead of a wrong number.
Frequently Asked Questions
What’s the difference between net area and total area under a curve?
Net area is the plain definite integral, where any part of the curve below the x-axis subtracts from the total. Total area treats every part as positive, which is usually what “find the area under the curve” is asking for. This calculator shows both so you can see the difference.
How do I find the bounds for the area between two curves?
Set the two functions equal to each other and solve for x — those solutions are the curves’ intersection points, and typically the a and b you want. This calculator does not find those points for you; enter them once you’ve solved for them or read them off a graph.
Does the order of f(x) and g(x) matter for area between two curves?
No. Because the calculator integrates the absolute value of the difference, |f(x) − g(x)|, it gives the same enclosed area regardless of which function you enter first.
Can this calculator handle trig, exponential, or logarithmic functions?
Yes. It supports sin, cos, tan and their inverses, sinh/cosh/tanh, exp, ln, log (base 10), log2, sqrt, cbrt, abs, and the constants pi and e, in addition to ordinary polynomial expressions.
Why doesn’t this show the antiderivative or algebraic steps?
It computes the area numerically (Simpson’s Rule) rather than symbolically, which lets it handle functions that don’t have a clean antiderivative. That trade-off means it reports an accurate number rather than a term-by-term symbolic solution.
Can r(θ) be negative in the polar area formula?
The formula squares r(θ), so a negative value contributes the same area as its positive counterpart. Be aware that for curves where r changes sign within your angle range, the enclosed region can overlap itself, which is a property of the curve rather than a calculator limitation.