Answer
a) 418.76
b) .09 percent higher
Work Step by Step
We know that the energy needed is given by:
$Q = \int_0^{100} mc(T) dt$
Since the mass is one, this simplifies to:
$Q = \int_0^{100} c(T) dt$
Plugging c(T) into this equation, we find:
$Q=418,763.33 \ J \approx 418.76 \ kJ$
b) If we were to use the approximation, the value would be:
$Q = mc\Delta T = (1)(4184)(100)=418,400 \ J$
The actual value is $\fbox{.0867 percent higher}$
