a. Graph the functions f₁(x) = sin² x and f₂(x) = sin² 2x on the interval [0, π]. Find the area under these curves on [0, π].

b. Graph a few more of the functions f_n(x) = sin² nx on the interval [0, π], where n is a positive integer. Find the area under these curves on [0, π]. Comment on your observations.

c. Prove thathas the same value for all positive integers n.
d. Does the conclusion of part (c) hold if sine is replaced with cosine?
e. Repeat parts (a), (b), and (c) with sin² x replaced with sin⁴ x. Comment on your observations.

f. Challenge problem: Show that for m = 1, 2, 3, . . ,

Transcribed Image Text:

S"sin² (nx) dx 1·3•5. . · (2m – 1) TT sin2m x dx | cos" 2m х dх — п . 2•4•6•·2m 2m