The function and its inverse are continuous. If , find

I really have no idea how to proceed.

Answer: 5f(5)

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- October 20th 2010, 08:03 AMnvwxgnInverse function on definite integral
The function and its inverse are continuous. If , find

I really have no idea how to proceed.

Answer: 5f(5) - October 20th 2010, 08:48 AMAckbeet
I would interpret the problem geometrically, in terms of area. Obviously,

is the area under the function if we assume which we will do for now. Now the graph of the function is just the graph of , but flipped over the line So the integral

is the area under the function If you plot this up for a few functions, you might notice that if you flip the area represented by

over the line , that will just equal the area required to fill the rectangle whose area is

The result follows for positive functions, at least.

I should point out that one assumption here, that is justified by the hypothesis that both and are continuous, is that exists. That means would be 1-1. - October 20th 2010, 08:56 AMTheEmptySet
Hint in the 2nd integral let

and the new limits of integration are

then use integration by parts.

Also this can be seen on a graph.

The first integral is in blue the inverse is in brown.

Attachment 19399

Edit: Wow I am really slow at posting today