The Area Between Curves Calculator computes the area enclosed between two polynomial functions over an interval using definite integration. Handles intersections and sign changes — used in calculus, physics, economics, and any field where the area between functions is meaningful.
21.333333
sq units
21.333333
sq units
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21.333333
sq units
21.333333
sq units
1
0
4
Two curves cross each other and enclose a region. What is the area of that region? This is one of the most common applications of definite integration in calculus — and the calculator for area between curves computes it using numerical integration of the difference |f(x) − g(x)| over the specified interval, handling the sign changes that occur when the curves cross within the interval.
The area A between f(x) and g(x) over the interval [x₁, x₂] is:
A = ∫[x₁ to x₂] |f(x) − g(x)| dx
When f(x) ≥ g(x) throughout [x₁, x₂], this simplifies to:
A = ∫[x₁ to x₂] [f(x) − g(x)] dx
If the curves intersect within the interval, the integrand changes sign and the absolute value must be respected — otherwise regions where g(x) > f(x) would subtract from the total area rather than add to it. This calculator detects intersections numerically and integrates each subregion with the correct sign. The definite integral calculator computes a single function's integral when area under a curve (not between two curves) is needed.
For a closed region bounded by two curves, the integration limits are typically the x-coordinates where f(x) = g(x). Setting the two functions equal and solving gives the intersection points. For f(x) = x² and g(x) = x + 2: x² = x + 2 → x² − x − 2 = 0 → (x−2)(x+1) = 0 → x = −1, 2. Area = ∫[−1 to 2] (x + 2 − x²) dx = [x²/2 + 2x − x³/3] from −1 to 2 = (2 + 4 − 8/3) − (1/2 − 2 + 1/3) = 10/3 − (−13/6) = 9/2 = 4.5 square units. When intersection points are not obvious analytically, numerical root-finding is required — which this calculator handles automatically.
Area between curves represents meaningful quantities across many disciplines:
The integral calculator and integrals category provide the complete calculus toolkit for single and multiple function integration.
A critical distinction in area between curves problems: the signed area counts regions where g > f as negative, while the unsigned (geometric) area counts all enclosed regions as positive. Most physical problems want unsigned area — total enclosed area regardless of which curve is on top. Mathematical problems sometimes ask for signed area to detect net displacement or net economic benefit. This calculator computes unsigned area by default, which is correct for "find the total area of the enclosed region" style problems.
The area between $$f(x) = a_1x^2 + b_1x + c_1$$ and $$g(x) = a_2x^2 + b_2x + c_2$$ over $$[a, b]$$ is:
$$\text{Area} = \int_a^b |f(x) - g(x)|\,dx = \int_a^b |Ax^2 + Bx + C|\,dx$$
where $$A = a_1 - a_2$$, $$B = b_1 - b_2$$, $$C = c_1 - c_2$$.
Signed integral (exact, using polynomial antiderivative):
$$\int_a^b (Ax^2 + Bx + C)\,dx = \frac{A}{3}(b^3 - a^3) + \frac{B}{2}(b^2 - a^2) + C(b - a)$$
Absolute area (numerical, using composite Simpson's rule with $$n = 100$$ subintervals):
$$\int_a^b |h(x)|\,dx \approx \sum_{k=0}^{9} S_k$$
where each $$S_k$$ is a Simpson's rule panel over 10 subintervals of width $$h = (b-a)/100$$. The Simpson pattern applies weights $$1, 4, 2, 4, 2, \ldots, 4, 1$$ and multiplies by $$h/3$$.
When the signed integral equals the absolute area, the curves do not cross in the interval. When they differ, the curves cross and the absolute area is larger.
The Area Between Curves is the total unsigned area enclosed between $$f(x)$$ and $$g(x)$$. This is always non-negative and represents the actual geometric region between the two curves, regardless of which is on top.
The Signed Integral is $$\int_a^b [f(x) - g(x)]\,dx$$ without absolute value. If positive, $$f(x)$$ is predominantly above $$g(x)$$; if negative, $$g(x)$$ is predominantly above. If the signed integral differs significantly from the absolute area, the curves cross within the interval.
The Difference Coefficients $$A$$, $$B$$, $$C$$ define the quadratic $$h(x) = Ax^2 + Bx + C = f(x) - g(x)$$. Setting $$h(x) = 0$$ and solving gives the x-coordinates where the curves intersect. The discriminant $$B^2 - 4AC$$ determines whether intersections exist.
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f(x) = -x^2+4 is a downward parabola with vertex at (0,4), crossing the x-axis at x = -2, 2. Since f(x) >= 0 on [-2,2], the signed and absolute integrals agree. Exact: integral of (4-x^2)dx from -2 to 2 = [4x - x^3/3] = (8 - 8/3) - (-8 + 8/3) = 32/3 = 10.6667.
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h(x) = x^2 - 2x = x(x-2), which is negative on [0,2] and positive on [2,3]. Signed integral = 9-9+0 - 0 = 0... Actually: integral of (x^2-2x) from 0 to 3 = [x^3/3 - x^2] from 0 to 3 = 9-9 = 0. But the absolute area = integral |x^2-2x| = area on [0,2] + area on [2,3] = 4/3 + 4/3 = ... The curves cross at x=2, creating two separate regions.
The signed integral $$\int [f(x)-g(x)]dx$$ can be positive or negative and allows cancellation between regions where $$f > g$$ and regions where $$g > f$$. The absolute area $$\int |f(x)-g(x)|dx$$ always counts every region as positive. When the two values agree, the curves do not cross. When they differ, crossings exist and the absolute area is the geometrically correct enclosed area.
Set $$f(x) = g(x)$$, which gives $$Ax^2 + Bx + C = 0$$ where $$A = a_1-a_2$$, $$B = b_1-b_2$$, $$C = c_1-c_2$$. Use the quadratic formula: $$x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}$$. The calculator displays the coefficients $$A$$, $$B$$, $$C$$ to help you compute intersections manually.
Yes. Set the $$x^2$$ coefficients to zero for either or both functions, and you get linear functions. The area between two lines is a trapezoid (or triangle if they share an endpoint), and the calculator handles this correctly.
The calculator integrates over whatever bounds you specify. If the curves intersect outside your bounds, the enclosed area within the bounds is still correctly computed. To find the total enclosed area between all intersection points, first find where the curves cross and set those as your bounds.
Simpson's rule with 100 subintervals provides very high accuracy for polynomial integrands. For quadratic difference functions $$|Ax^2+Bx+C|$$, the error is typically less than $$10^{-6}$$ for reasonable bound values. The only source of error is the absolute value making the integrand non-smooth at crossing points.
An area of zero means $$f(x) = g(x)$$ everywhere on the interval (the curves are identical). If the signed integral is zero but the absolute area is positive, it means equal portions of the interval have $$f > g$$ and $$g > f$$, creating symmetric regions that cancel in the signed integral.
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