Area Under Curve Calculator (Trapezoidal Rule for f(x) = x²)

This calculator approximates the area under the curve for the function f(x) = x² using the Trapezoidal Rule.

Start X Value (a):

End X Value (b):

Number of Trapezoids (n):

How to Calculate Area Under a Curve

The "area under a curve" is a fundamental concept in mathematics, particularly in calculus, with wide-ranging applications across various scientific and engineering disciplines. It refers to the area enclosed by a function's graph, the x-axis, and two vertical lines (representing the start and end points of an interval).

What Does Area Under a Curve Represent?

Conceptually, the area under a curve can represent different physical quantities depending on what the axes represent:

If the y-axis is velocity and the x-axis is time, the area under the curve represents the total distance traveled.
If the y-axis is force and the x-axis is displacement, the area represents the work done.
In probability, the area under a probability density function over an interval gives the probability of an event occurring within that interval.
In economics, it can represent total cost, total revenue, or consumer/producer surplus.

Mathematically, finding the area under a curve is equivalent to evaluating a definite integral of the function over a given interval.

Methods for Calculating Area Under a Curve

There are primarily two approaches to calculate the area under a curve:

1. Analytical Method (Definite Integral)

For many continuous functions, the exact area can be found using the fundamental theorem of calculus. This involves finding the antiderivative (or indefinite integral) of the function and then evaluating it at the upper and lower limits of the interval. If F(x) is the antiderivative of f(x), then the area from a to b is F(b) - F(a).

While precise, this method requires knowledge of calculus and the ability to find the antiderivative, which isn't always straightforward or even possible for all functions.

2. Numerical Integration Methods

When an analytical solution is difficult or impossible, or when you only have discrete data points, numerical integration methods provide an approximation of the area. These methods involve dividing the area under the curve into a series of simpler shapes (like rectangles or trapezoids) whose areas can be easily calculated and summed up.

Common numerical methods include:

Riemann Sums: Approximating the area with rectangles.
Trapezoidal Rule: Approximating the area with trapezoids.
Simpson's Rule: Approximating the area with parabolas (more accurate than trapezoids for smooth curves).

Understanding the Trapezoidal Rule

The calculator above uses the Trapezoidal Rule, a popular and relatively simple numerical integration technique. Instead of using rectangles (as in Riemann sums), the Trapezoidal Rule approximates the area under the curve by dividing the interval into a number of smaller sub-intervals and forming trapezoids under the curve within each sub-interval.

Here's how it works:

Divide the Interval: The interval [a, b] is divided into n equal sub-intervals.
Calculate Width (h): The width of each sub-interval (and thus each trapezoid) is h = (b - a) / n.
Form Trapezoids: For each sub-interval [x_i, x_{i+1}], a trapezoid is formed by connecting the points (x_i, f(x_i)) and (x_{i+1}, f(x_{i+1})) with a straight line. The parallel sides of the trapezoid are f(x_i) and f(x_{i+1}), and its height is h.
Area of a Trapezoid: The area of a single trapezoid is (1/2) * (base1 + base2) * height = (1/2) * (f(x_i) + f(x_{i+1})) * h.
Sum the Areas: The total approximate area under the curve is the sum of the areas of all these trapezoids.

The formula for the Trapezoidal Rule is:

Area ≈ (h/2) * [f(x₀) + 2f(x₁) + 2f(x₂) + … + 2f(x_n-1) + f(x_n)]

Where:

a is the start X value.
b is the end X value.
n is the number of trapezoids (sub-intervals).
h = (b - a) / n is the width of each sub-interval.
x₀ = a, x₁ = a + h, ..., x_n = b are the x-coordinates of the sub-interval endpoints.

Using the Area Under Curve Calculator

Our calculator specifically approximates the area under the curve for the function f(x) = x². To use it:

Start X Value (a): Enter the lower limit of the interval for which you want to calculate the area.
End X Value (b): Enter the upper limit of the interval.
Number of Trapezoids (n): Enter the number of sub-intervals you want to divide the area into. A higher number of trapezoids generally leads to a more accurate approximation.
Click "Calculate Area" to see the result.

Example Calculations:

Example 1: Area under f(x) = x² from x=0 to x=2 with 4 trapezoids

Start X Value (a): 0
End X Value (b): 2
Number of Trapezoids (n): 4
h = (2 - 0) / 4 = 0.5
x-values: 0, 0.5, 1, 1.5, 2
f(x) values: f(0)=0, f(0.5)=0.25, f(1)=1, f(1.5)=2.25, f(2)=4
Area ≈ (0.5/2) * [f(0) + 2f(0.5) + 2f(1) + 2f(1.5) + f(2)]
Area ≈ 0.25 * [0 + 2(0.25) + 2(1) + 2(2.25) + 4]
Area ≈ 0.25 * [0 + 0.5 + 2 + 4.5 + 4]
Area ≈ 0.25 * [11] = 2.75

(The exact analytical answer for ∫x² dx from 0 to 2 is [x³/3] from 0 to 2 = 8/3 ≈ 2.666667. Our approximation is close!)

Example 2: Area under f(x) = x² from x=1 to x=3 with 10 trapezoids

Start X Value (a): 1
End X Value (b): 3
Number of Trapezoids (n): 10
h = (3 - 1) / 10 = 0.2
The calculator will sum up the areas of 10 trapezoids.
Using the calculator with these inputs will yield approximately 8.68.

(The exact analytical answer for ∫x² dx from 1 to 3 is [x³/3] from 1 to 3 = (27/3) – (1/3) = 9 – 1/3 = 26/3 ≈ 8.666667. Again, the approximation is very close.)

Accuracy and Limitations

The accuracy of the Trapezoidal Rule (and most numerical integration methods) improves as the number of sub-intervals (n) increases. More trapezoids mean smaller h values, leading to a closer fit to the curve and thus a more accurate approximation of the area.

While powerful, numerical methods provide approximations, not exact values (unless the function itself is linear, in which case trapezoids are exact). For functions with high curvature, more trapezoids are needed to achieve a good approximation.

How to Calculate Area Under Curve