2021 AMC 10A Problems/Problem 19
Problem 19
The area of the region bounded by the graph ofis
, where
and
are integers. What is
?
Solution 1
In order to attack this problem, we need to consider casework:
Case 1:
Substituting and simplifying, we have , i.e.
, which gives us a circle of radius
centered at
.
Case 2:
Substituting and simplifying again, we have , i.e.
. This gives us a circle of radius
centered at
.
Case 3:
Doing the same process as before, we have , i.e.
. This gives us a circle of radius
centered at
.
Case 4:
One last time: we have , i.e.
. This gives us a circle of radius
centered at
.
After combining all the cases and drawing them on the Cartesian Plane, this is what the diagram looks like:
Now, the area of the shaded region is just a square with side length
with four semicircles of radius
.
The area is
. The answer is
which is
Solution by Bryguy
Diagram anonymous (?): https://artofproblemsolving.com/wiki/index.php/File:Image_2021-02-11_111327.png (someone please help link file thanks)
Video Solution (Using absolute value properties to graph)
~ pi_is_3.14