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Question Number 2163 by Filup last updated on 06/Nov/15

For : y = f (x) \to x = g (y)

Therefore :

{\begin{cases} x (t) = t \\ y (t) = f (t) \end{cases}

let r (t) = ⟨ x (t), y (t) ⟩

∴ \int r d t = ⟨ \int t d t, \int f (t) d t ⟩

Does :

\int x (t) d t = \int g (y) d y

and

\int y (t) d t = \int f (x) d x

Commented by prakash jain last updated on 06/Nov/15

g (y) = x = t

y = f (x) = f (t)

d y = f^{'} (t) d t

\int x (t) d t = \int t d t

\int g (y) d y = \int t f^{'} (t) d t \neq \int t d t

Answered by prakash jain last updated on 06/Nov/15

Second part

\int y (t) d t = \int f (x) d x is correct since y (t) = f (t)

\int f (t) d t = \int f (x) d x ∵ x = t

Commented by Filup last updated on 06/Nov/15

T h a n k s

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