The intersection of the cycloid line and the X axis is the origin, the intersection is A, and the section from A to the origin on the cycloid line is L, so the area S =1/2 ∫ (OA+L) xdy-ydx =1/2 ∫ (OA) xdy-ydx+1.
Curve integral is a kind of integral. The value of the integral function is not along the interval, but along a specific curve, which is called the integral path. There are many kinds of curve integrals. When the integration path is a closed curve, it is called loop integration or contour integration. Curve integral can be divided into the first category and the second category.
Extended data:
The difference between two kinds of curve integrals mainly lies in the difference of integral elements; The integral element of arc-length curve integral is arc-length element ds; For example, the curve integral of l ∫f(x, y)*ds.
The integral element of the curve integral of the coordinate axis is the coordinate element dx or dy, such as the curve integral of L' ∫P(x, y)dx+Q(x, y)dy. But the curve integral of arc length is usually positive because of its physical meaning, while the curve integral of coordinate axis can get different symbols according to different paths.
In curve integration, the integrand can be a scalar function or a vector function. The value of the integral is the riemann sum of the function value of each point on the path multiplied by the corresponding weight (generally arc length, when the integral function is a vector function, it is generally the scalar product of the function value and the infinitesimal vector of the curve).
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