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Area Between Two Curves Calculator

Enter the expressions of the curves, set the limits, and select the variables. The free Area Between Two Curves Calculator will compute the enclosed area over the given interval using definite integrals.

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Use this calculator to find the area between two curves over a given interval. It computes the total area by evaluating the definite integral of the difference between the two functions.

What is Area Between Two Curves?

In mathematics, the area between two curves is the region bounded by two functions \(f(x)\) and \(g(x)\) over an interval \([a, b]\). If \(f(x) \ge g(x)\) on \([a,b]\), the area is given by:

Formula:

\[ A = \int_a^b \big(f(x) - g(x)\big)\, dx \]

An online integral calculator can help evaluate these integrals quickly.

Steps to Calculate the Area Between Curves

Identify the two curves: \(y = f(x)\) and \(y = g(x)\).
Find their intersection points by solving \(f(x) = g(x)\).
Set up the integral \(A = \int_a^b [f(x) - g(x)] dx\) with the intersection points as limits.
Evaluate the definite integral to find the area.

Example:

Find the area between the curves \(x^2 + 4y - x = 2\) and the line \(x = y\).

Rewrite the first curve as a function of \(y\):
\(x^2 + 4y - x = 2 \Rightarrow 4y = -x^2 + x + 2 \Rightarrow y = \frac{-x^2 + x + 2}{4}\)
Intersection points with \(y = x\):
\(\frac{-x^2 + x + 2}{4} = x \Rightarrow -x^2 - 3x + 2 = 0 \Rightarrow x = -1, 2\)
Define \(f(x)\) and \(g(x)\):
\(f(x) = \frac{-x^2 + x + 2}{4}, \quad g(x) = x\)
Set up the integral:
\[ A = \int_{-1}^{2} \left[f(x) - g(x)\right] dx = \int_{-1}^{2} \left(\frac{-x^2 + x + 2}{4} - x\right) dx \]
Compute the integral to find the area (use a calculator for exact value).

Area Between Two Curves

How the Calculator Works

Input:

Enter the two expressions of the curves (in terms of \(x\) or \(y\)).
Provide the upper and lower limits of integration.
Click "Calculate" to find the area.

Output:

Shows your input equations clearly.
Displays the area between the curves along with step-by-step integral calculations.

FAQs

Can the area be negative?

If \(f(x) < g(x)\) over an interval, the integral is negative. The signed value represents the area direction, but the magnitude gives the actual area.

Why use definite integrals?

Definite integrals provide the numerical area between curves for given limits. Indefinite integrals only give the family of functions.

Difference between definite and indefinite integrals:

Indefinite integral: yields a function plus a constant (\(C\)).
Definite integral: gives a numerical value over a specific interval.

EndNote

The area between two curves represents the magnitude of the region bounded by the curves. Using an online calculator simplifies the process and avoids errors in manual calculation.

References

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