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In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. [1] The terms path integral , curve integral , and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane .
In mathematics, the definite integral ()is the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the lines x = a and x = b, such that area above the x-axis adds to the total, and that below the x-axis subtracts from the total.
The gradient theorem implies that line integrals through gradient fields are path-independent. In physics this theorem is one of the ways of defining a conservative force . By placing φ as potential, ∇ φ is a conservative field .
Integration is the basic operation in integral calculus.While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful.
Finally, as t goes from 1 to +∞, the point follows the part of the circle in the second quadrant from (0, 1) to (−1, 0). Here is another geometric point of view. Draw the unit circle, and let P be the point (−1, 0). A line through P (except the vertical line) is determined by its slope.
This visualization also explains why integration by parts may help find the integral of an inverse function f −1 (x) when the integral of the function f(x) is known. Indeed, the functions x(y) and y(x) are inverses, and the integral ∫ x dy may be calculated as above from knowing the integral ∫ y dx.
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related to: line integrals practice problems examples worksheet 1 freeThis site is a teacher's paradise! - The Bender Bunch