Definite Integrals
A definite integral is a mathematical operation that allows us to calculate the area under a curve between two points on an interval. It gives us a precise value for the total area between the curve and the x-axis.
The definite integral is denoted by ∫, which represents the integral symbol, and has a lower and upper limit that defines the interval of integration. The lower limit is usually denoted by a and the upper limit by b.
The formula for calculating the definite integral of a function f(x) from a to b is:
∫[a to b] f(x) dx
where f(x) is the integrand, dx represents an infinitely small change in x, and the integral symbol ∫ indicates that we are summing up all these infinitely small changes in x over the interval [a, b].
To compute the definite integral, we can use various methods, such as the fundamental theorem of calculus, the power rule, substitution, or integration by parts. These methods allow us to find an antiderivative of the integrand, which is a function whose derivative gives the original integrand.
The result of a definite integral is a single value, which represents the area between the curve and the x-axis over the given interval. This value can be positive or negative, depending on the shape of the curve and the orientation of the interval. A positive value indicates that the area is above the x-axis, while a negative value indicates that the area is below the x-axis.
Definite integrals are used in various fields, such as physics, economics, and engineering, to calculate quantities such as work, displacement, area, volume, and average values.
Overall, definite integrals are a powerful tool for calculating the area under curves and finding precise values of quantities related to functions. They provide a fundamental concept in calculus and are widely used in various mathematical and scientific applications.