An integral is a fundamental concept in calculus that represents the area under a curve or the accumulation of a quantity over a given interval. Integrals come in two main types: definite and indefinite integrals.
A definite integral calculates the net area under a curve between two specified points, while an indefinite integral represents an antiderivative of a function, which, when differentiated, yields the original function.
Integrals are essential for solving various problems in mathematics, physics, and engineering. They can be used to calculate areas, volumes, and even to find the average value of a function over a given interval. In addition, integrals play a vital role in the study of differential equations and the formulation of the Fundamental Theorem of Calculus, which links differentiation and integration.
Integration techniques include substitution, integration by parts, partial fractions, and trigonometric substitution. Furthermore, numerical methods such as the Riemann sum, trapezoidal rule, and Simpson’s rule can be employed to approximate definite integrals when finding an exact solution is challenging or impossible.