Solution 73: Schröder Path Pattern Avoidance

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• Published on Jan 6, 2019
• We find Schröder paths that avoid UDUDUD, UDUUDD, and HHHHHH by enumerating the desired Dyck paths.
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• Tuanicular 5 months ago

Can you make a solution video to this problem, I’m stumped by it and can’t seem to understand the given solution, so I hope you can show me the process.
Given f(x) = x^5 - x^3 + 4x, find the integral from 0 to 4 of f^-1(x) wrt x
Greatly appreciate if you upload a video on this, and always love your vids!

• dolphin lunggrin 3 months ago

An easy way would be using the fact that the graph of f^-1 is just the graph of f(x) mirrored at the line y=x. or in other words you get the inverse by swapping the x and y axis.
Since f(0)=0 and f(1)=4, the graph of f is a curve from (0,0) to (1,4). Now sketch the graph between these points and draw a rectangle with the corners (0,0), (0,4), (1,4), (1,0). the upper left part of it is the integral you want and the bottom right part is the integral from 0 to 1 of f(x). The area of the rectangle and the integral of f are easy to calculate and with those you get your solution by taking the difference of the two.
area of the rectangle is 4
integral of f is x^6/6 - x^4/4 + 2x^2 and with the bounds 0, 1 you get 23/12
so the integral from 0 to 4 of f^-1(x) = 4 - 23/12 = 25/12 without ever calculating what the inverse of f even is much less integrating it.

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• Nerdiconium 5 months ago +1

wait, what about one U after the string of D's?

• LetsSolveMathProblems  5 months ago +7

A Dyck path (or a Schröder path) cannot end with a U because its terminal point must be on the x-axis.

• Kwekinator117 5 months ago +4

How do you get good at solving these problems?

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