Alright, I'm nearly done this piece of crappy calculus, but I have no idea how to properly evaluate definite integrals because even though the homework I'm doing requires it... We haven't done it in class yet?

Trying to find the area between two sine waves, f(x) = 11.6164sin(0.85x-4.99)+23.8656, and 8.2264sin(0.883x-4.77)+17.3568, basically what do I do next to evaluate the infinite series?



My text book is shaky at best.

Um i'm not sure exactly what you're asking for... could be that I forgot all my calculus/series from like forever ago.
I'm assuming you would just find the area (solve the anti derivative) with the limits 9 and 0 whereas normally when solving indefinite integrals you add the "+ C" at the end to indicate an unknown constant (and sometimes you would have to solve for this value of C.)

∫^9_0 ( 11.6164 * sin( 0.85x - 4.99 ) + 23.8656 )dx - ∫^9_0 ( 8.2264 * sin(0.883x - 4.77) + 17.3568 )dx
= ∫^9_0 (11.6164 * sin( 0.85x - 4.99 ))dx + ∫^9_0 (23.8656)dx - ∫^9_0 (8.2264 * sin(0.883x - 4.77))dx - ∫^9_0 (17.3568)dx
= -11.6164 * cos(0.85x - 4.99) / 0.85 |^9_0 + 23.8656x |^9_0 + 8.2264 * cos(0.883x - 4.77) / 0.883 |^9_0 - 17.3568x |^9_0
= -13.67cos(0.85x - 4.99) + 9.32cos(0.883x - 4.77) + 6.51x |^9_0
= -13.67cos(0.85*9 - 4.99) + 9.32cos(0.883*9 - 4.77) + 6.51*9 - (-13.67cos(0 - 4.99) + 9.32cos(0 - 4.77) + 0)
= 64.6

someone may have to verify and make sure I made no errors...

I don't ever remember having to try to evaluate infinite Riemann sums without an integral, and I'm not sure how to go about it without proving the Fundamental Theorem..

the integral though is

[-13.666cos(0.85x-4.99) + 23.8656x ₀|⁹ - [-9.313cos(0.883x-4.77) + 17.3568x ₀|⁹
= [-13.666cos(0.85x-4.99) + 9.313cos(0.883x-4.77) + 6.5088x ₀|⁹

Oh, if MA does the numbers I'll check it, one sec.

Yes, 64.593 (with the 3-digits-past standard)

where f(x) _a|^b denotes f(b) - f(a)

Post has been edited 1 time(s), last time on Mar 9 2009, 2:44 am by DT_Battlekruser.

Crap, I accidently was using the wrong equations anyways XD Thanks anyways, I should be able to do it now.