Expert: Steve Holleran - 9/25/2007

Question
find the  finite region in the postive  quadrant  bounded  by the  curve  y =cos^3x and the curve y= sin^3x and the  y axis
My problem is  finding  the points  of  intersection between the  two   functions .... i dont know how to solve sinx= cosx  far less cos^3x+sin^3x.

Thanks in advance  
Andy

Answer
Hi Andrew,

Okay, so you have cos^3 x = sin^3 x .  First, subtract sin^3 x from both sides to get:

           cos^3 x - sin^3 x = 0

now factor this as a difference of cubes
(Remember, the factoring pattern is

 a^3 - b^3 = (a - b)(a^2 + ab + b^2))

and you should get:

   cos x - sin x)(cos^2 x + cos x sin x + sin^2 x) = 0

The second factor gives no real solutions, so the only solution is when cos x - sin x = 0 , or

                cos x = sin x

             cos x / cos x = sin x / cos x

                      1 = tan x   so x = pi/4 (45 degrees)

                                       = .7853

So then your integration is

 INT(from 0 to .7853) [(sin^3 x - cos^3 x) dx]

The only way I know to evaluate this is using a graphing calculator or other computer software.  I got 0.5118 as the area.

Hope this helps a little
Steve