Calculate Area Under Curve

Understanding the Area Under a Curve

The area under a curve is a fundamental concept in calculus with wide-ranging applications across various scientific and engineering disciplines. Geometrically, it represents the region bounded by the function's graph, the x-axis, and two vertical lines (the lower and upper bounds). This area can signify different physical quantities depending on the context of the function. For instance, if the curve represents velocity over time, the area under it gives the total displacement. If it represents a force over distance, the area gives the work done.

Calculating Area Under a Quadratic Function

This calculator specifically helps you determine the definite integral, or the area under the curve, for a quadratic function of the form f(x) = Ax2 + Bx + C. This type of function creates a parabolic curve, and finding the area under it involves applying the principles of definite integration.

The definite integral of a function f(x) from a lower bound a to an upper bound b is given by ∫ab f(x) dx. For our quadratic function, the antiderivative (or indefinite integral) is:

F(x) = (A/3)x3 + (B/2)x2 + Cx + K (where K is the constant of integration, which cancels out in definite integrals).

To find the definite integral, we evaluate the antiderivative at the upper bound and subtract its value at the lower bound:

Area = F(b) - F(a)

Area = [(A/3)b3 + (B/2)b2 + Cb] - [(A/3)a3 + (B/2)a2 + Ca]

How to Use the Calculator

Enter the coefficients A, B, and C for your quadratic function f(x) = Ax2 + Bx + C. Then, specify the lower and upper bounds of the interval over which you want to calculate the area. The calculator will instantly provide the exact area under the curve for that specified range.

Examples

Let's look at a few examples:

  1. Simple Parabola: Calculate the area under f(x) = x2 from x=0 to x=3.
    • A = 1, B = 0, C = 0
    • Lower Bound = 0, Upper Bound = 3
    • Calculation: F(x) = (1/3)x3. Area = F(3) – F(0) = (1/3)(3)3 – (1/3)(0)3 = 9 – 0 = 9.
  2. Parabola with Offset: Calculate the area under f(x) = 2x2 + 3x + 1 from x=1 to x=2.
    • A = 2, B = 3, C = 1
    • Lower Bound = 1, Upper Bound = 2
    • Calculation: F(x) = (2/3)x3 + (3/2)x2 + x. F(2) = (2/3)(8) + (3/2)(4) + 2 = 16/3 + 6 + 2 = 16/3 + 8 = 16/3 + 24/3 = 40/3. F(1) = (2/3)(1) + (3/2)(1) + 1 = 2/3 + 3/2 + 1 = 4/6 + 9/6 + 6/6 = 19/6. Area = 40/3 – 19/6 = 80/6 – 19/6 = 61/6 ≈ 10.1667.
  3. Negative Area: Calculate the area under f(x) = x2 - 4 from x=0 to x=1.
    • A = 1, B = 0, C = -4
    • Lower Bound = 0, Upper Bound = 1
    • Calculation: F(x) = (1/3)x3 – 4x. F(1) = (1/3)(1) – 4(1) = 1/3 – 4 = 1/3 – 12/3 = -11/3. F(0) = 0. Area = -11/3 – 0 = -11/3 ≈ -3.6667. (A negative area indicates the curve is below the x-axis in that interval).

Limitations and Further Exploration

This calculator is designed for quadratic functions. For more complex functions (e.g., trigonometric, exponential, or higher-order polynomials), or for functions defined by discrete data points, numerical integration methods like the Trapezoidal Rule or Simpson's Rule are often employed. These methods approximate the area by dividing the region into smaller, simpler shapes (trapezoids or parabolas) and summing their areas.

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