Area Under Graph Calculator

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Expertly Reviewed by David Chen, CFA — Specialist in Quantitative Analysis & Mathematics

Calculating the definite integral of a function provides the exact area between the graph and the x-axis. Use this professional **Area Under Graph Calculator** to solve complex integration problems instantly, featuring step-by-step breakdowns for students and engineers.

Area Under Graph Calculator

Use ‘x’ as variable. Supported: +, -, *, /, ^, sin, cos, exp, log.

The Area Under the Curve is:

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Area Under Graph Calculator Formula:

Area = ∫ab f(x) dx

Source: Wolfram MathWorld – Definite Integrals

Variables:

  • f(x): The mathematical function representing the curve.
  • a: The starting point on the x-axis (Lower Boundary).
  • b: The ending point on the x-axis (Upper Boundary).
  • dx: The infinitesimal width of the rectangular slices.

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What is Area Under Graph Calculator?

The area under a graph represents the accumulation of quantities. In geometry, this is straightforward for shapes like rectangles or triangles, but for curved functions, calculus is required. Our calculator uses numerical integration techniques to find the total area bounded by a function $f(x)$ and the x-axis between two vertical lines $x = a$ and $x = b$.

This tool is essential for calculating displacement from velocity-time graphs, work done from force-displacement graphs, or probability densities in statistics.

How to Calculate Area Under Graph (Example):

Suppose you want to find the area under $f(x) = x^2$ from $x = 0$ to $x = 3$.

  1. Identify the integral: Set up the expression ∫03 x2 dx.
  2. Find the anti-derivative: The integral of $x^2$ is $(1/3)x^3$.
  3. Apply Boundaries: Calculate [(1/3)(3)3] – [(1/3)(0)3].
  4. Result: (1/3)(27) – 0 = 9 square units.
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Frequently Asked Questions (FAQ):

  • Can the area be negative? Yes, if the graph is below the x-axis, the definite integral results in a negative value.
  • What is Simpson’s Rule? It is a numerical method for integration that approximates the area using parabolic segments.
  • What does “dx” mean? It represents a very small change in $x$, indicating the direction of integration.
  • Do I need to be in Radian mode? Yes, all trigonometric functions in this calculator assume input $x$ is in Radians.
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