Find The Area Of The Region Calculator

Easily calculate the area of the region bounded by two functions, y=f(x) and y=g(x), over a specified interval [a, b] using our Area Between Curves Calculator.

Calculate Area Between Curves

Formula Used (Simpson's Rule for Area between f(x) and g(x)): Area ≈ (h/3) * [ (f(x₀)-g(x₀)) + 4(f(x₁)-g(x₁)) + 2(f(x₂)-g(x₂)) + … + 4(f(xₙ₋₁)-g(xₙ₋₁)) + (f(xₙ)-g(xₙ)) ], where h = (b-a)/n and n is even.

Sampled Values

x	f(x)	g(x)	f(x) – g(x)
Enter values and calculate to see data.

Table showing f(x), g(x), and their difference at sample points within [a, b].

What is an Area Between Curves Calculator?

An Area Between Curves Calculator is a tool used to determine the area of the region enclosed between two functions, say y = f(x) and y = g(x), over a specified interval [a, b] on the x-axis. This area is found by calculating the definite integral of the absolute difference between the two functions, |f(x) – g(x)|, from x=a to x=b. If one function is consistently above the other (e.g., f(x) ≥ g(x)) in the interval, the area is the integral of (f(x) – g(x)) dx from a to b.

This calculator is particularly useful for students of calculus, engineers, physicists, and anyone needing to find the area bounded by curves without performing manual integration, especially when the functions are complex or when numerical methods are preferred. Our Area Between Curves Calculator uses numerical integration (Simpson's rule) for accuracy.

Common misconceptions include thinking the area is simply the difference between the integrals of f(x) and g(x) without considering which function is greater, or that it only works for simple polynomial functions. This Area Between Curves Calculator can handle a wide range of functions input using JavaScript's Math object.

Area Between Curves Formula and Mathematical Explanation

The area A of the region bounded by the curves y = f(x) and y = g(x) between x = a and x = b, where f(x) ≥ g(x) for all x in [a, b], is given by the definite integral:

A = ∫_a^b [f(x) – g(x)] dx

If the curves intersect within the interval [a, b], or it's unknown which function is greater, the area is found by integrating the absolute difference:

A = ∫_a^b |f(x) – g(x)| dx

Our Area Between Curves Calculator assumes you input f(x) as the upper curve and g(x) as the lower curve within the interval for simplicity in the basic formula display, but it effectively calculates ∫(f(x)-g(x))dx. For accurate results where curves cross, one might need to split the integral at intersection points.

Since symbolic integration of arbitrary functions is complex, our calculator uses Simpson's Rule for numerical integration, which approximates the definite integral. For an even number of intervals 'n', Simpson's Rule is:

∫_a^b h(x) dx ≈ (Δx/3) [h(x₀) + 4h(x₁) + 2h(x₂) + 4h(x₃) + … + 4h(xₙ₋₁) + h(xₙ)]

where h(x) = f(x) – g(x), Δx = (b-a)/n, and xᵢ = a + i*Δx.

Variables Table

Variable	Meaning	Unit	Typical Range
f(x)	The upper function	Expression	Math expressions (e.g., x*x, Math.sin(x))
g(x)	The lower function	Expression	Math expressions (e.g., x, 0)
a	Lower limit of integration	Number	Any real number
b	Upper limit of integration	Number	Any real number (b > a)
n	Number of intervals (for Simpson's rule)	Integer	Even integer ≥ 2
Δx or h	Width of each interval ((b-a)/n)	Number	Positive real number

Practical Examples

Example 1: Area between y = x² and y = x from 0 to 1

• f(x) = x (upper curve in [0,1])
• g(x) = x*x (lower curve in [0,1])
• a = 0
• b = 1
• n = 100

Using the Area Between Curves Calculator with f(x)="x", g(x)="x*x", a=0, b=1, n=100, we find the area ≈ 0.166667. The exact integral of (x – x²) from 0 to 1 is [x²/2 – x³/3] from 0 to 1 = (1/2 – 1/3) – 0 = 1/6 ≈ 0.166667.

Example 2: Area between y = sin(x) and y = cos(x) from 0 to π/4

• f(x) = Math.cos(x) (upper curve in [0, π/4])
• g(x) = Math.sin(x) (lower curve in [0, π/4])
• a = 0
• b = Math.PI/4 (approx 0.7854)
• n = 100

Using the calculator with f(x)="Math.cos(x)", g(x)="Math.sin(x)", a=0, b=Math.PI/4, n=100, the area ≈ 0.414213. The exact integral of (cos(x) – sin(x)) from 0 to π/4 is [sin(x) + cos(x)] from 0 to π/4 = (sin(π/4)+cos(π/4)) – (sin(0)+cos(0)) = (√2/2 + √2/2) – (0+1) = √2 – 1 ≈ 0.41421356.

How to Use This Area Between Curves Calculator

1. Enter Upper Function f(x): Input the mathematical expression for the upper function f(x) in terms of 'x'. Use JavaScript's Math object for functions like `Math.sin(x)`, `Math.pow(x,2)` (or `x*x`), `Math.exp(x)`, `Math.log(x)`.
2. Enter Lower Function g(x): Input the expression for the lower function g(x). If finding the area under f(x) down to the x-axis, g(x) is 0.
3. Enter Lower Limit (a): Input the starting x-value of your interval.
4. Enter Upper Limit (b): Input the ending x-value of your interval (ensure b > a).
5. Enter Number of Intervals (n): Choose an even number of intervals for Simpson's rule. More intervals generally mean higher accuracy but take slightly longer. 100 is a good starting point.
6. Calculate: Click "Calculate Area". The Area Between Curves Calculator will display the result, intermediate values, and update the chart and table.
7. Read Results: The primary result is the calculated area. Intermediate values show interval width, intervals used, and functions. The chart visualizes the area, and the table shows sample points.

Key Factors That Affect Area Between Curves Results

• The Functions f(x) and g(x): The shapes of the curves directly define the region and its area. Complex functions can lead to complex regions.
• The Limits of Integration (a, b): The interval [a, b] defines the horizontal extent of the region whose area is being calculated. Changing 'a' or 'b' changes the area.
• The Number of Intervals (n): For numerical integration, 'n' affects the accuracy. A larger 'n' gives a better approximation of the true area but increases computation.
• Intersection Points: If f(x) and g(x) cross within [a, b], the "upper" and "lower" functions switch. To get the total area, you might need to integrate |f(x)-g(x)| or split the integral at intersections if calculating manually. Our calculator finds ∫(f(x)-g(x))dx, so ensure f(x) is generally above g(x) or interpret negative results carefully.
• Correct Function Syntax: Using incorrect mathematical syntax for f(x) or g(x) will lead to errors or incorrect results. Use `*` for multiplication, `Math.pow(base, exp)` or `**` for powers, and `Math.` prefix for standard functions.
• Interval Width (Δx or h): Derived from (b-a)/n, smaller interval widths (from larger n) lead to more accurate numerical integration.

Frequently Asked Questions (FAQ)

What if g(x) is above f(x) in the interval?

The calculator computes the integral of f(x) – g(x). If g(x) > f(x), the result will be negative, representing the negative of the area between g(x) (upper) and f(x) (lower). To find the geometric area, take the absolute value or swap f(x) and g(x).

How accurate is the Area Between Curves Calculator?

The accuracy depends on the number of intervals 'n' used for Simpson's rule and the behavior of the functions. For smooth functions, a larger 'n' (e.g., 1000 or more) gives very accurate results.

Can I use this calculator for area under a single curve?

Yes, to find the area under f(x) down to the x-axis, set g(x) = 0.

What if the functions f(x) or g(x) are not defined everywhere in [a, b]?

The functions must be defined and continuous (or piecewise continuous) across the interval [a, b] for the integral to be well-defined in the standard sense. The calculator might produce errors or NaN if it encounters undefined values (e.g., division by zero, log of zero).

How do I input constants like π or e?

Use `Math.PI` for π and `Math.E` for e.

What if the curves intersect between a and b?

Our calculator calculates ∫[f(x)-g(x)]dx. If they cross, f(x)-g(x) changes sign. To find the total geometric area, you should identify intersection points c (a < c < b) and calculate ∫|f(x)-g(x)|dx, which might involve splitting the integral: ∫_a^c|f(x)-g(x)|dx + ∫_c^b|f(x)-g(x)|dx. Or ensure f(x) is always the upper function input and g(x) the lower if you know their relationship.

Why does the number of intervals have to be even?

Simpson's rule, the numerical method used, requires an even number of intervals 'n' to group them into pairs for parabolic approximation.

What does NaN mean in the result?

NaN (Not a Number) means the calculation resulted in an undefined value. This can happen if your functions involve operations like division by zero, square root of a negative number, or log of zero/negative numbers at some point in the interval, or if the function syntax is incorrect.