What is a line integral?

A line integral is a type of mathematical integral where the function being integrated is evaluated along a curve instead of an interval. The function can be either a scalar field or a vector field, and the integral sums the field values along the path, weighted by curve characteristics like arc length or a differential vector. These integrals provide continuous analogues for physical concepts, such as calculating the work done on an object moving through a force field.

What we know

• A line integral is an integral where the function to be integrated is evaluated along a curve. [2]
• The function integrated in a line integral may be a scalar field or a vector field. [2]
• The value of a line integral is the sum of field values on the curve weighted by arc length or the scalar product of the vector field with a differential vector. [2]
• A line integral generalizes standard integration by replacing an interval with a curve connecting two points in space for functions of two or more variables. [3]

Where to go next

• how it workedHow is work calculated using line integrals in physics?
• the bigger pictureWhat is a surface integral?
• compared withHow does the Riemann integral define integration on an interval?
• a case in pointWhat is a contour integral in the complex plane?
• whenHow did Isaac Newton and Gottfried Wilhelm Leibniz formulate integration?
• an unexpected connectionHow do continuous line integrals relate to discrete sums?An integral is fundamentally the continuous analog of a sum, translating discrete additions into continuous path evaluations.

Sources

• [1]line integral (Wikidata Q467699) · CC0 1.0
• [2]Line integral · CC BY-SA 4.0
• [3]Integral · CC BY-SA 4.0
• [4]Henstock–Kurzweil integral · CC BY-SA 4.0
• [5]Riemann integral · CC BY-SA 4.0