"In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations."
Finding areas, volumes, work done, etc.
Area between curves: Finding the area of a region bounded by two or more curves using integration.
Volume of solids of revolution: Finding the volume of a three-dimensional object generated by rotating a two-dimensional shape about an axis.
Arc length: Finding the length of a curve in the Cartesian plane using integration.
Surface area: Finding the surface area of a three-dimensional object using integration.
Work: Calculating the amount of work done by a force moving an object along a curve in the Cartesian plane.
Center of mass: Finding the center of mass of a two-dimensional object using integration.
Moments of inertia: Calculating the moment of inertia of an object with respect to an axis using integration.
Differential equations: Using integration to solve differential equations and model real-world phenomena.
Taylor series: Approximating functions using infinite series generated by integration.
Polar coordinates: Converting Cartesian equations to polar form and using integration to find area, arc length, and volume of objects in polar coordinates.
Improper integrals: Evaluating integrals with one or both limits being infinite or the integrand having a singularity.
Applications to physics: Using integration in physics, such as calculating gravitational attraction between objects or modeling heat transfer.
Applications to economics: Using integration in economics, such as estimating consumer surplus or producer surplus.
Applications to engineering: Using integration in engineering, such as calculating stress and strain in materials or modeling fluid flow.
Area: The definite integral is used to find the area bounded by a curve and the x-axis or y-axis.
Volume: The definite integral is used to find the volume of solids of revolution, such as cylinders, cones, and spheres.
Work: The definite integral is used to find the work done by a force along a path.
Fluid pressure: The definite integral is used to find the pressure exerted by a fluid on a surface.
Probability: The definite integral is used to find the probability of an event occurring.
Center of mass: The definite integral is used to find the center of mass of a system.
Arc-length: The definite integral is used to find the arc-length of a curve.
Moment of Inertia: The definite integral is used to find the moment of inertia of a system.
Electric charge: The definite integral is used to find the electric charge in a system.
Thermal conductivity: The definite integral is used to find the thermal conductivity of a material.
Magnetic field: The definite integral is used to find the magnetic field produced by a current-carrying conductor.
Radioactivity: The definite integral is used to find the rate of decay of a radioactive substance.
Population dynamics: The definite integral is used to model and study the growth or decline of populations.
Optimal control: The definite integral is used to find the optimal control strategy for systems.
Signal processing: The definite integral is used to analyze signals and filter them for noise reduction.
"Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation."
"Integration started as a method to solve problems in mathematics and physics, such as finding the area under a curve or determining displacement from velocity."
"The integrals enumerated here are called definite integrals, which can be interpreted as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line."
"Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative."
"Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function; in this case, they are also called indefinite integrals."
"The fundamental theorem of calculus relates definite integrals with differentiation and provides a method to compute the definite integral of a function when its antiderivative is known; differentiation and integration are inverse operations."
"Although methods of calculating areas and volumes dated from ancient Greek mathematics, the principles of integration were formulated independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century."
"Isaac Newton and Gottfried Wilhelm Leibniz... thought of the area under a curve as an infinite sum of rectangles of infinitesimal width."
"Bernhard Riemann later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into infinitesimally thin vertical slabs."
"In the early 20th century, Henri Lebesgue generalized Riemann's formulation by introducing what is now referred to as the Lebesgue integral; it is more robust than Riemann's in the sense that a wider class of functions are Lebesgue-integrable."
"Integrals may be generalized depending on the type of the function as well as the domain over which the integration is performed."
"A line integral is defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting the two endpoints of the interval."
"In a surface integral, the curve is replaced by a piece of a surface in three-dimensional space."
"Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation."
"Integration started as a method to solve problems in mathematics and physics, such as finding the area under a curve or determining displacement from velocity."
"It is more robust than Riemann's in the sense that a wider class of functions are Lebesgue-integrable."
"Bernhard Riemann later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into infinitesimally thin vertical slabs."
"Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative."
"The fundamental theorem of calculus relates definite integrals with differentiation and provides a method to compute the definite integral of a function when its antiderivative is known; differentiation and integration are inverse operations."