Fundamental theorem of calculus
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Apibrėžimas
amžius 16 lygis-ang. 3
If ∫
f(x)dx exists, and if there is a function F(x) which has [(
dF)/(
dx)]
(x) =
f(x) in the range
a ≤
x ≤
b, then
| | ⌠
⌡ | b
a
| f(x)dx = F(b) − F(a) |
|
ie, differentiation is the inverse of integration.
Apibrėžimas
amžius 19 lygis-ang. 4
Real case: let F be continuously differentiable on [a,b] , with derivative F
′=f . Then ∫
ab f exists, and
Complex case: let D be a domain and f : D →
C be continuous. If f has an antiderivative F and ϕ: [a,b] → D is a path, then
| | ⌠
⌡ |
ϕ
| f(z) dz = F( ϕ(b))−F( ϕ(a)). |
|
Finansuojamas: EU Socrates Minerva, HeyMath!, Cambridge University Press