Area Between Two Curves Calculator
Calculate the area between two functions using definite integration. Our calculator handles polynomial, trigonometric, and exponential functions with step-by-step solutions, multiple integration methods, and detailed mathematical explanations.
Last updated: December 15, 2024
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Area Calculation Result
Area Between Curves:
From 0 to 2 using analytical method
Functions:
x^2x[0, 2]Integration Setup:
Solution Steps:
- Step 1: Set up the integral ∫[0, 2] |f(x) - g(x)| dx
- Step 2: Identify upper function: x^2
- Step 3: Identify lower function: x
- Step 4: Check for intersection points in the interval
- Step 5: Evaluate the definite integral using analytical
- Step 6: Calculate the absolute area between curves
Verification:
Verification: Area = 0.6667 square units ✓
Quick Example Result
For functions f(x) = x² and g(x) = x over interval [0, 2]:
Area = 0.6667 square units
How This Calculator Works
Our area between curves calculator uses definite integration to find the region bounded by two functions. The process involves setting up the integral with proper bounds, determining which function is upper or lower, and applying integration techniques to calculate the exact or approximate area between the curves.
The Integration Formula
Area = ∫ᵃᵇ |f(x) - g(x)| dxWhere f(x) and g(x) are the two functions, and [a,b] is the interval of integration. The absolute value ensures we always get a positive area regardless of which function is on top.
Shows the shaded region between two curves over the integration interval
Mathematical Foundation
The area between curves is a fundamental application of definite integration from the Fundamental Theorem of Calculus. When two continuous functions bound a region, the area can be calculated by integrating the absolute difference between the functions over the specified interval. This method works for any integrable functions and provides exact results when analytical integration is possible.
- Determine which function is upper and which is lower in each subinterval
- Split the integral at intersection points to maintain proper orientation
- Apply appropriate integration techniques (substitution, parts, etc.)
- Sum the areas of all subintervals for the total area
Sources & References
- Stewart Calculus - Applications of Integration ChapterComprehensive coverage of area between curves methods
- MIT OpenCourseWare - Single Variable Calculus IntegrationDetailed examples and problem-solving techniques
- Khan Academy - Area Between Curves Tutorial SeriesStep-by-step video explanations and practice problems
Need help with other calculus calculations? Check out our derivative calculator and concavity calculator.
Get Custom Calculator for Your BusinessExample Calculation
Given Functions:
- Upper Function: f(x) = x²
- Lower Function: g(x) = x
- Lower Limit: a = 0
- Upper Limit: b = 2
Solution Steps:
- Set up: ∫₀² |x² - x| dx
- Determine x² ≥ x when x ≥ 1
- Split: ∫₀¹ (x - x²) dx + ∫₁² (x² - x) dx
- Integrate and evaluate: 0.6667 square units
Result: Area = 0.6667 square units
This represents the total area of the region bounded by the two curves over the given interval.
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