Fundamental Theorem of Calculus/Second Part/Also presented as
Jump to navigation
Jump to search
Theorem
The Fundamental Theorem of Calculus can also be presented in the following form:
Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ expressed as a normal first order ODE:
- $(1): \quad \dfrac {\d y} {\d x} = \map f x$
Let $c \in \R$ be an arbitrary real number.
Then there exists a unique solution $\map F x$ to $(1)$ on $\closedint a b$ such that $\map F a = c$, given by the definite integral:
- $\ds \map F x = c + \int_a^x \map f t \rd t$
Sources
- 1978: Garrett Birkhoff and Gian-Carlo Rota: Ordinary Differential Equations (3rd ed.) ... (previous) ... (next): Chapter $1$ First-Order Differential Equations: $2$ Fundamental Theorem of the Calculus