Let y = f(x) be the function defined on -1 <= x <= 1 by the formula

\[y = \sqrt{4 - (x - 1)^2} - 1.\]

This is a graph of y = f(x):

If a graph of x = f(y) is overlaid on the graph above, then one fully enclosed region is formed by the two graphs. What is the area of that region, rounded to the nearest hundredth?

Guest Apr 7, 2021

#1**+1 **

$y=\sqrt{4-(x-1)^2}-1$

$(y+1)^2=4-(x-1)^2$

$(x-1)^2=4-(y+1)^2$

$x=1+\sqrt{4-(y+1)^2}$

Switch $x,y$ and get $y=1+\sqrt{4-(x+1)^2}$.

Graph and you will see magic.

thedudemanguyperson Apr 7, 2021