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Category: Arc Length

Taylor series sinx

Taylor series sinx

Input interpretation
series | sin(x)
Series expansion at x=0
x – x^3/6 + x^5/120 + O(x^7)
(Taylor series)
Approximations about x=0 up to order 3
(order n approximation shown with n dots)
Antiderivative of e^x^2

Antiderivative of e^x^2

Indefinite integral
integral e^(x^2) dx = 1/2 sqrt(pi) erfi(x) + constant
Alternate form of the integral
e^(x^2) F(x) + constant
Series expansion of the integral at x=0
x + x^3/3 + x^5/10 + O(x^6)
(Taylor series)
Series expansion of the integral at x=∞
1/2 (e^(x^2) (1/x + 1/(2 x^3) + 3/(4 x^5) + O((1/x)^6)) – i sqrt(pi))
Antiderivative of cos

Antiderivative of cos

Indefinite integral
integral cos(x) dx = sin(x) + constant
Alternate form of the integral
1/2 i e^(-i x) – 1/2 i e^(i x) + constant
Series expansion of the integral at x=0
x – x^3/6 + x^5/120 + O(x^6)
(Taylor series)
Definite integral
integral_0^(pi/2) cos(x) dx = 1
Definite integral mean square
integral_0^(2 pi) (cos^2(x))/(2 pi) dx = 1/2 = 0.5
Antiderivative of sin

Antiderivative of sin

Indefinite integral
integral sin(x) dx = -cos(x) + constant
Alternate form of the integral
-1/2 e^(-i x) – e^(i x)/2 + constant
Series expansion of the integral at x=0
-1 + x^2/2 – x^4/24 + O(x^6)
(Taylor series)
Definite integral over a half-period
integral_0^pi sin(x) dx = 2
Definite integral mean square
integral_0^(2 pi) (sin^2(x))/(2 pi) dx = 1/2 = 0.5
Length of parametric curve

Length of parametric curve

Input interpretation
parametric plane curves | arc length
Result
astroid | s = 6 a
bifolium | s = 1/6 a (sqrt(4 sqrt(2) – 5) (4 sqrt(2) (K(1/7 (-57 – 40 sqrt(2))) + F(sin^(-1)(1/7 (9 – 4 sqrt(2)))|1/7 (-57 – 40 sqrt(2)))) + 5 E(1/7 (-57 – 40 sqrt(2))) + 5 E(sin^(-1)(1/7 (9 – 4 sqrt(2)))|1/7 (-57 – 40 sqrt(2)))) + 8/7 (8 + 5 sqrt(2)))
cardioid | s = 8 a
Cayley sextic | s = 6 pi a
cycloid of Ceva | s = a (-3 K(13/16) + 16 E(13/16) + 3 Pi(1/4|13/16))
circle | s = 2 pi a
circle parallel curve | s = 2 pi (a + k)
circular arc | s = a p
cornoid | s = 4 a (-3 K(-2) + E(-2) + 3 Pi(1/4|-2))
deltoid | s = (16 a)/3
Example plots
Length of a parametric curve

Length of a parametric curve

Input interpretation
parametric plane curves | arc length
Result
astroid | s = 6 a
bifolium | s = 1/6 a (sqrt(4 sqrt(2) – 5) (4 sqrt(2) (K(1/7 (-57 – 40 sqrt(2))) + F(sin^(-1)(1/7 (9 – 4 sqrt(2)))|1/7 (-57 – 40 sqrt(2)))) + 5 E(1/7 (-57 – 40 sqrt(2))) + 5 E(sin^(-1)(1/7 (9 – 4 sqrt(2)))|1/7 (-57 – 40 sqrt(2)))) + 8/7 (8 + 5 sqrt(2)))
cardioid | s = 8 a
Cayley sextic | s = 6 pi a
cycloid of Ceva | s = a (-3 K(13/16) + 16 E(13/16) + 3 Pi(1/4|13/16))
circle | s = 2 pi a
circle parallel curve | s = 2 pi (a + k)
circular arc | s = a p
cornoid | s = 4 a (-3 K(-2) + E(-2) + 3 Pi(1/4|-2))
deltoid | s = (16 a)/3
Example plots