H2.4-6a Problem Notes

The derivative d/dt can be pulled through an integral sign, provided some 
technical conditions hold. So, generally expect that 

   d/dt)(integral) = integral with (d/dt) moved onto the integrand.

The fundamental theorem of calculus says 

		integral of f'(x) from x=a to x=b  EQUALS  f(b) - f(a)

If f(x) = (d/dx) u(x,t) with t held fixed, then f'(x) = (d/dx)^2 u(x,t) 
and the fundamental theorem says

		integral of (d/dx)^2 u(x,t) from x=a to x=b EQUALS f(b) - f(a)
                where
                f(b) = (du/dx)(b,t) = u_x(b,t) and f(a) = (du/dx)(a,t) = u_x(a,t)


Limits taken through an integral sign are possible with technical conditions satisfied.
The basic results try to make legal the formula

	limit of the integral of f_n  =  integral of the limit of f_n

For example, limit of 1 + (1/n) sin(x) = 1 as n tends to infinity. Then

        limit of integral of 1 + (1/n) sin(x)  EQUALS the integral of 1