The purpose of the integral in mathematics is to take the infinitely small areas under (or inside) a curve to calculate the area under (or inside of) it. Each line under a curve has a specific length, determined by the definition of the function. Integrals take the area of each line (a value whose limit is zero, since it's one-dimensional) and adds them together until the entire area under the curve (and only that area) is filled with an infinite number of lines.
Integration is useful for finding the exact area under (or inside of) the curve. Another method of finding the area uses a technique called a Riemann sum, which requires no integration, but provides only an estimate. The accuracy of that estimate depends on how many iterations are used in the Riemann sum.
When dealing with very small, iterative quantities, you can use a recursive summation to calculate the total. If the quantities are small enough, you can use an integral to accomplish the same task, usually with less work. These summations can be described as areas.