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Area Under The Curve Calculator

Enter the function, limits, and select the variables. The calculator will determine the area under the bell curve, providing detailed calculations.

Enter a Function:

Lower Limit: (inf = ∞ , pi = π)

Upper Limit: (inf = ∞ , pi = π)

W.R.T

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Area Under the Curve Calculator

The Area Under the Curve (AUC) Calculator helps you evaluate definite integrals quickly and accurately. By entering a function and specifying upper and lower limits, you can compute the exact area under the curve along with detailed step-by-step integration.

What Is Area Under the Curve (AUC)?

In calculus, the area under a curve represents the definite integral of a function between two limits. If we have a function f(x) defined between x = a and x = b, the area is calculated as:

AUC = ∫_a^b f(x) dx

If the function lies above the x-axis, the integral is positive. If it lies below the x-axis, the integral becomes negative, but the geometric area is always considered positive.

Area Under the Curve Graph

Area Under the Curve Formula

AUC = ∫_a^b f(x) dx

a = lower limit
b = upper limit
f(x) = function

Example 1

Find the area under: f(x) = 6x + 3, from x = 0 to x = 4

Step 1: Set up the integral

∫₀⁴ (6x + 3) dx

Step 2: Integrate

∫ 6x dx = 3x²
∫ 3 dx = 3x

Step 3: Antiderivative

F(x) = 3x² + 3x

Step 4: Apply limits

F(4) − F(0)

= (3·4² + 3·4)

= 60

Final Answer: AUC = 60

Example 2

Find the area under: y = x³ + 5, from x = 0 to x = 1

∫₀¹ (x³ + 5) dx

∫ x³ dx = x⁴/4
∫ 5 dx = 5x

F(x) = x⁴/4 + 5x

F(1) − F(0) = 21/4

Final Answer: AUC = 21/4

How the Calculator Works

Input

Enter the function f(x)
Enter lower and upper limits
Click Calculate

Output

Exact value of the definite integral
Step-by-step solution

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