Area Under the Curve Calculator
Use this calculator to approximate the definite integral of a function over a given interval using the Trapezoidal Rule.
Use 'x' as the variable. For math functions, use 'Math.sin(x)', 'Math.cos(x)', 'Math.exp(x)', 'Math.log(x)', 'Math.pow(x, y)'. You can also use 'Math.PI' for pi.
More subintervals generally lead to a more accurate approximation.
Understanding the Area Under the Curve
The "area under the curve" is a fundamental concept in calculus, representing the definite integral of a function over a specified interval. Geometrically, it's the area bounded by the function's graph, the x-axis, and the vertical lines at the start and end points of the interval. If the function dips below the x-axis, the area in that region is considered negative, reflecting the net change rather than absolute area.
Why is it Important?
The area under the curve has vast applications across various fields:
- Physics: The area under a velocity-time graph gives displacement. The area under a force-distance graph gives work done.
- Economics: It can represent total cost, total revenue, or consumer/producer surplus.
- Probability: The area under a probability density function over an interval gives the probability of an event occurring within that interval.
- Engineering: Used in signal processing, fluid dynamics, and structural analysis.
- Statistics: Crucial for understanding cumulative distribution functions and hypothesis testing.
How This Calculator Works: The Trapezoidal Rule
Since finding the exact analytical solution for every function can be complex or impossible, numerical methods are often used to approximate the area. This calculator employs the Trapezoidal Rule, a simple yet effective method for numerical integration.
The Trapezoidal Rule works by dividing the area under the curve into a series of trapezoids instead of rectangles (as in Riemann sums). For each small subinterval, it connects the function's values at the endpoints with a straight line, forming a trapezoid. The sum of the areas of all these trapezoids provides an approximation of the total area under the curve.
The formula for the Trapezoidal Rule is:
Area ≈ (h/2) * [f(a) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(b)]
Where:
his the width of each subinterval, calculated as(b - a) / n.ais the lower bound of the integration.bis the upper bound of the integration.nis the number of subintervals.f(x)is the function whose area you want to calculate.
The more subintervals (larger 'n') you use, the narrower each trapezoid becomes, and generally, the more accurate the approximation will be.
How to Use the Calculator
- Function f(x): Enter your mathematical function. Use 'x' as the variable. For standard mathematical functions like sine, cosine, exponential, or logarithm, use JavaScript's
Mathobject (e.g.,Math.sin(x),Math.exp(x),Math.log(x),Math.pow(x, 2)for x squared). You can also useMath.PIfor the value of pi. - Lower Bound (a): Enter the starting x-value of your interval.
- Upper Bound (b): Enter the ending x-value of your interval.
- Number of Subintervals (n): Specify how many trapezoids you want to use for the approximation. A higher number generally yields better accuracy.
- Click "Calculate Area" to see the result.
Examples
Example 1: Area under f(x) = x² from 0 to 1
- Function f(x):
x*x - Lower Bound (a):
0 - Upper Bound (b):
1 - Number of Subintervals (n):
100 - Expected Result (exact): 1/3 or approximately 0.333333
- Calculator Result: Approximately 0.333333 (will vary slightly based on 'n')
Example 2: Area under f(x) = sin(x) from 0 to π
- Function f(x):
Math.sin(x) - Lower Bound (a):
0 - Upper Bound (b):
Math.PI(approximately 3.14159) - Number of Subintervals (n):
1000 - Expected Result (exact): 2
- Calculator Result: Approximately 2.000000 (will vary slightly based on 'n')
Example 3: Area under f(x) = 2x + 3 from -1 to 2
- Function f(x):
2*x + 3 - Lower Bound (a):
-1 - Upper Bound (b):
2 - Number of Subintervals (n):
50 - Expected Result (exact): 12 (Area of a trapezoid or integral of 2x+3 from -1 to 2)
- Calculator Result: Approximately 12.000000