How does analysis help calculate the area inside an irregular curve?
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr2. As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr2—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.
In analysis, calculating the area inside an irregular curve is primarily achieved through the process of integration. Because shapes with curved edges are difficult to measure directly, analysis employs limiting processes to approximate the area using simpler regions.123
The Riemann Integral
The most common approach, the Riemann integral, involves slicing the region under a curve into thin vertical strips. By approximating the area of these strips with rectangles, mathematicians can calculate the sum of these areas. As the thickness of these slices tends toward zero, the sum converges to a specific value, which is defined as the integral of the function.3
The Fundamental Theorem of Calculus
The fundamental theorem of calculus provides a powerful shortcut by establishing that integration is the inverse of differentiation. If a function represents the area under a curve , then the derivative of that area function is the original curve itself (). This allows for the calculation of areas by finding the antiderivative of the function.12
Advanced Methods
For functions that are too irregular to be represented pictorially or handled by Riemann sums, the Lebesgue integral is used. Instead of partitioning the -axis, it partitions the -values and uses the concept of measure to determine the area, offering a more general and robust framework for integration.4