Area Under The Curve

TL;DR

• Computes the total area enclosed by a curve on a graph over a specified interval of x.
• Practically approximated by summing areas of many thin rectangles; the exact value is the limit as the rectangle widths → 0.
• Used to convert a curve (function) and bounds into a single numeric area value.

Definition

Area Under the Curve, also known as the definite integral, is a mathematical concept used to calculate the area enclosed by a curve on a graph. The definite integral is defined as the limit of the sum of the areas of rectangles as the number of segments approaches infinity:

$$\int_a^b f(x)\,dx = \lim_{n\to\infty} \sum (\text{width of each rectangle}) \times (\text{height of each rectangle})$$

Explanation

• To approximate the area under a curve, divide the region under the curve into small vertical rectangles.
• For each rectangle compute area = width × height, where the height comes from the function value at or within the rectangle.
• As the number of rectangles increases and their widths decrease, the sum of rectangle areas approaches the definite integral.
• Calculus provides the definite integral as the limit of these sums, making exact area calculation feasible without summing many finite rectangles.
• Determining the definite integral requires the function f(x) that defines the curve and the bounds (lower and upper limits) a and b for x.

Examples

Straight line with slope 1

Consider a straight line on a graph with coordinates (x, y) and slope 1 (for every unit increase in x, y increases by 1). The area under the line can be approximated by dividing the region into small rectangles, computing each rectangle’s area using length × width, and summing these areas.

Parabola y = x^2

For the curve defined by y = x^2 (coordinates (x, x^2)), the curve is parabolic. To calculate the area under this curve, divide it into small rectangles; their widths vary because the curve is not straight. The slope of the curve at each point can be calculated using the derivative of the curve’s equation to determine the segment behavior. Summing the areas of all rectangles and taking the limit as the number of rectangles approaches infinity yields the definite integral. For example:

$$\int_0^2 x^2\,dx = \lim_{n\to\infty} \sum (\text{width of each rectangle}) \times (\text{height of each rectangle})$$

Use cases

• Physics
• Engineering
• Economics

Notes or pitfalls

• Calculating the area by summing finite rectangles can be tedious and time-consuming for complex curves with many segments.
• The definite integral replaces this tedious process by taking the limit as the number of rectangles approaches infinity.
• You must determine the bounds of the integral (the lower and upper limits of x) and the function f(x) that defines the curve before computing the definite integral.

Related terms

• Definite integral
• Derivative
• Calculus