Area Under Curve Calculator

Last updated: 19 May 2026 | Author: Brij | Review By: Irshad

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Calculate the area under a curve using definite integrals, lower and upper limits, and numerical approximation methods. Easy area under curve calculator for calculus, mathematics, engineering, and physics applications.

Function Equation f(x):

Enter function using x as variable. Examples: x^2, sin(x), 2*x+3, x^3, sqrt(x), e^x

Lower Limit (a): Start point

Upper Limit (b): End point

Number of Intervals: Higher = more accuracy

Numerical Method: Integration method

Area Under Curve Results

Definite Integral:

∫ f(x) dx

Mathematical expression

Area Under Curve:

sq units

Total area

Approximate Numerical Area:

sq units

Numerical approximation

f(a) = f(x₀):

value

Function at lower limit

f(b) = f(xₙ):

value

Function at upper limit

Average Value:

units

Average f(x) over interval

Step Size (Δx):

units

Width of each interval

Graph Result:

Graph of f(x) with shaded area under curve

An Area Under Curve Calculator helps calculate the area between a mathematical curve and the x-axis over a specified interval. This concept is widely used in calculus, mathematics, engineering, physics, statistics, and economics.

By entering the function equation, lower limit, upper limit, and number of intervals, you can estimate the definite integral and total area under the curve.

Definite Integral Formula

The area under a curve is calculated using a definite integral.

$\text{Area}=\int_a^b f(x)\,dx$

Where:

$f(x)$ = Function equation
$a$ = Lower limit
$b$ = Upper limit

Interval Width Formula

For numerical approximation methods, interval width is:

$\Delta x=\frac{b-a}{n}$

Where:

$n$ = Number of intervals

Trapezoidal Approximation Formula

One common approximation method is the trapezoidal rule.

$\text{Area}\approx\frac{\Delta x}{2}\left[f(x_0)+2f(x_1)+\cdots+2f(x_{n-1})+f(x_n)\right]$

How the Area Under Curve Calculator Works

The calculator evaluates the function between the lower and upper limits using integration techniques. If intervals are provided, numerical approximation methods such as the trapezoidal rule may also be used.

This calculation is useful for:

calculus problems
physics applications
probability distributions
engineering analysis
economics and statistics

Example Calculation

Suppose:

Function Equation: $f(x)=x^2$

Lower Limit: $a=0$

Upper Limit: $b=3$

Number of Intervals: $n=6$

Step 1: Write the Definite Integral

$\text{Area}=\int_0^3 x^2\,dx$

Step 2: Integrate the Function

$\int x^2\,dx=\frac{x^3}{3}$

Step 3: Apply the Limits

$\left[\frac{x^3}{3}\right]_0^3$

Step 4: Calculate Final Area

$\frac{3^3}{3}-\frac{0^3}{3}$

Final Result

$\text{Area}=9$

Applications of Area Under Curve Calculation

This calculator is useful for:

Calculus and mathematics
Engineering analysis
Physics calculations
Probability and statistics
Economics and finance
Scientific modeling

Benefits of Using an Area Under Curve Calculator

Fast and accurate integration
Simplifies calculus problems
Useful for students and engineers
Supports numerical approximation methods
Reduces manual calculation errors

FAQs

What is the area under a curve?

The area under a curve represents the accumulated quantity between the curve and the x-axis over a given interval.

What is a definite integral?

A definite integral calculates the exact area between two limits.

Why are intervals used?

Intervals are used in numerical approximation methods such as the trapezoidal rule.

Can this calculator solve trigonometric functions?

Yes, functions such as:

$\sin(x)$
$\cos(x)$
$e^x$
polynomials

can be evaluated.

What is the trapezoidal rule?

The trapezoidal rule is a numerical method used to estimate the area under a curve.

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