Definite Integrals: Area Under Curve Simulator & Guide
A definite integral of f from a to b gives the signed area between the curve and the x-axis. By the Fundamental Theorem of Calculus you evaluate it as F(b) minus F(a), where F is any antiderivative of f.
When you take the integral of a function, you're really just trying to find the area underneath the line.
For example, if you have a linear function that goes diagonally up, the area below that line forms a triangle. Technically, you don't even need calculus to find that area—you can just use simple geometry by multiplying half of the base by the height.
But what if the line is no longer straight and simple?
This is where integration comes in:
• Handling Irregular Shapes: When the curve is wavy or curved, standard geometry formulas fail. We use integrals as a mathematical tool to calculate the exact area of these irregular shapes.
• The Opposite of Derivatives: Integration is the exact opposite of differentiation. While taking a derivative finds the instantaneous slope, taking the integral finds the accumulated area under the graph itself.
How do you evaluate a definite integral?
Definite integral worked example: area under 3x squared
Evaluate the definite integral of f(x) = 3x² from x = 1 to x = 3.
1. Find the antiderivative: F(x) = ∫ 3x² dx = x³.
2. Evaluate at the limits of integration:
3. Compute the final value:
Another worked example
Evaluate the integral of 2x from x = 0 to x = 3. Step 1: an antiderivative of 2x is x squared. Step 2: evaluate at the bounds as 3 squared minus 0 squared. Step 3: the result is 9 minus 0, which equals 9.
Aligned with the College Board AP Calculus CED — Unit 6 (Integration and Accumulation of Change), Topic 6.7, with the area interpretation of Topic 6.6. College Board ↗
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