Tuesday
June 18, 2013

# Homework Help: definite integrals

Posted by .. on Thursday, April 26, 2012 at 11:44am.

integrate [sinx + cosx] from 0 to 2pie, where [ ] represents greatest integer function .

• definite integrals - Reiny, Thursday, April 26, 2012 at 12:42pm

∫ (sinx + cosx) from x = 0 to 2π
= -cosx + sinx | from 0 to 2π
= -cos(2π) + sin(2π) - (-cos0 + sin0)
= -1+ 0-(-1 + 0)
= 0

ahh, something is not right.
I know.....
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
http://www.wolframalpha.com/input/?i=sin%28x%29+%2B+cos%28x%29
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)

Related Questions

Trig........ - I need to prove that the following is true. Thanks (cosx / 1-sinx...
Trigonometry Check - Simplify #3: [cosx-sin(90-x)sinx]/[cosx-cos(180-x)tan​...
Pre-Calc - Trigonometric Identities Prove: (tanx + secx -1)/(tanx - secx + 1)= ...
Math - Verify the identity . (cscX-cotX)^2=1-cosX/1+cosX _______ sorry i cant ...
Math - dz =(&#8722;sinx + 2xy^2)dx +(2x^2 y)dy Integrate the differential to...
Pre calc - Find solutions on interval 0,2pie (cosx/1+sinx) + (1+sinx/cosx) = -4
Trigonometry. - ( tanx/1-cotx )+ (cotx/1-tanx)= (1+secxcscx) Good one! ...
math - Integrate sinx/sinx-cosx..... Plz note sinx-cosx as a whole is in the ...
Math - Pre- Clac - Prove that each of these equations is an identity. A) (1 + ...
Calculus (Definite Integrals) - How many definite integrals would be required to...