I Wish I could Be Your Integral
Geometry
Level
4
Which of the following curve encloses the largest area?
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1 solution
x 2 + y 2 = 1 . . . . . . . . . . ( 1 ) x 4 + y 4 = 1 . . . . . . . . . . ( 2 ) x 6 + y 6 = 1 . . . . . . . . . . ( 3 ) x 8 + y 8 = 1 . . . . . . . . . . ( 4 ) x 2 n + y 2 n = 1 . . . . . . . . . . a n d . . . . . . . . . . x 2 n + 2 m + y 2 n + 2 m = 1 . . . . . . . . . . n , m a n y + t i v e i n t e g e r . G i v e n x y a = 2 n 1 − x 2 n . . . . . . . . . . . ( a ) . . . . . . . . . . . . . y b = 2 n + 2 m 1 − x 2 n + 2 m . . . . . . . . . . ( b ) . S i n c e x ≤ 1 , e x c e p t a t t w o p o i n t s . 1 − x 2 n + 2 m > 1 − x 2 n . f o r a l l b u t t w o p o i n t s . A g a i n s i n c e t h e v a l u e u n d e r r a d i c a l s i g n i s l e s s t h a n o n e , h i g h e r r o o t g i v e s b i g g e r v a l u e . ( b ) h a s i t s r o o t b i g g e r b y 2 m s o a g a i n y b > y a . S o f o r e v e r y x , ( b ) h a s b i g g e r y . ∴ ( 4 ) w i l l h a v e t h e b i g g e s t a r e a .
We can plot the curves and verify that above is correct.
Note that through out the curve, only at two points all have same value. All other infinite points, the curve with higher power has bigger values.