Posted by Ethan on Wednesday, September 28, 2011 at 12:00am.
given the graph of f(x) = x sinx, 0<=x<=2pi
assuming that a quantity y changes at a rate of y' = xsinx, find by how much it will increase or decrease over 3pi/2 <= x <= 2pi

calculus  Steve, Wednesday, September 28, 2011 at 10:46am
This is just a problem in integrals. We have y', and we want to add up all the small changes given by that function.
Int[x sinx dx] can be solved using integration by parts.
Recall the product rule of derivatives:
(uv)' = u'v + uv'
so
uv' = (uv)'  u'v
Going the other direction,
Int[u dv] = uv  Int[v du]
So, we want to split up the integrand into two factors, where one part (u) gets simpler after differentiation, and the other part (dv) can be easily integrated.
Here, we have x sinx
If u = x, then du = dx
If dv = sinx, then v = cosx
Int[x sinx dx] = uv  Int[v du]
= x cosx  Int[cos x * dx]
Now we have a simple integrand, cosx
Int cos x dx = sin x
So, the final integration is
Int[x sinx dx] = uv  Int[v du]
= x cosx  Int[cos x * dx]
= x cosx + sin x
Evaluating at 3pi/2 and 2pi, we have
(2pi * 1 + 0)  (3pi/2 * 0 + 1) = 2pi + 3pi/2 + 1 = 1 + 7pi/2
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