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A parabolic segment is the region bounded by a parabola and line. To find the area of a parabolic segment, Archimedes considers a certain inscribed triangle. The base of this triangle is the given chord of the parabola, and the third vertex is the point on the parabola such that the tangent to the parabola at that point is parallel to the chord.
If D is a simple type of region with its boundary consisting of the curves C 1, C 2, C 3, C 4, half of Green's theorem can be demonstrated. The following is a proof of half of the theorem for the simplified area D , a type I region where C 1 and C 3 are curves connected by vertical lines (possibly of zero length).
The theorem stated above can be generalized. The circle γ can be replaced by any closed rectifiable curve in U which has winding number one about a. Moreover, as for the Cauchy integral theorem, it is sufficient to require that f be holomorphic in the open region enclosed by the path and continuous on its closure.
In differential geometry, the sign of the area of a region of a surface is associated with the orientation of the surface. [7] The area of a set A in differential geometry is obtained as an integration of a density : μ ( A ) = ∫ A d x ∧ d y , {\displaystyle \mu (A)=\int _{A}dx\wedge dy,} where d x and d y are differential 1-forms that make ...
The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). [43]
Let R denote the region enclosed by a curve r(φ) and the rays φ = a and φ = b, where 0 < b − a ≤ 2 π. Then, the area of R is [()]. The region R is approximated by n sectors (here, n = 5). A planimeter, which mechanically computes polar integrals
The Jordan curve (drawn in black) divides the plane into an "interior" region (light blue) and an "exterior" region (pink). In topology , the Jordan curve theorem ( JCT ), formulated by Camille Jordan in 1887, asserts that every Jordan curve (a plane simple closed curve) divides the plane into an " interior " region bounded by the curve and an ...
Take a to be greater than 1, so that the imaginary unit i is enclosed within the curve. The contour integral is ∫ C e i t z z 2 + 1 d z . {\displaystyle \int _{C}{\frac {e^{itz}}{z^{2}+1}}\,dz.} Since e itz is an entire function (having no singularities at any point in the complex plane), this function has singularities only where the ...