Posted by .. on Thursday, April 26, 2012 at 11:44am.
∫ (sinx + cosx) from x = 0 to 2π
= -cosx + sinx | from 0 to 2π
= -cos(2π) + sin(2π) - (-cos0 + sin0)
= -1+ 0-(-1 + 0)
ahh, something is not right.
when graphed some of the "area" is above and some is below the x-axis, so some of this stuff canceled.
So we have to look at the graph of sinx + cosx
looking at Wolfram's graph
shows that we have 2 x-intercepts
sinx + cosx = 0
sinx = -cosx
divide both sides by cos
sinx/cosx = -1
tanx = -1
x = 135° or 315° or (3π/4 or 7π/4)
so we need to do this in 3 parts
∫(sinx + cosx) dx from 0 to 3π/4 + ∫(-sinx - cosx) dx from 3π/4 to 7π/4 + ∫(sinx+cosx) dx from 7π/4 to 2π
guess what, I will let let you do all that beautiful arithmetic.
(Careful when you adjust that second integral)
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