Why is Area Under a Curve the Integral?

Area under a curve in integration represents the accumulation of quantities, such as distance, probability, or any variable quantity, over a continuous interval.

Answer: The area under a curve is represented by the integral because integration sums up an infinite number of infinitesimal elements to calculate the total area accurately.

When dealing with curved shapes or functions, it’s often impossible to find their area using simple geometric formulas. Integration, a fundamental concept in calculus, allows us to divide the curve into tiny intervals, approximate each interval as a rectangle, and then sum these rectangles to estimate the total area. As we make these intervals smaller and smaller, the sum of these rectangles approaches the actual area under the curve.

The integral symbol (∫) represents this summation process.

Hence, by finding the integral of a function over a specific interval, we are essentially finding the accumulated area between the curve and a reference axis within that interval.