$$
\begin{array}{|l|c|c|c|c|c|}
\hline
\mbox{ t (minutes)} & 0 & 4 & 9 & 15 & 20 \\
\mbox{ W(t) (degrees Fahrenheight)} & 55.0 & 57.1 & 61.8 & 67.9 & 71.0 \\
\hline
\end{array}
$$The Temperature of water in tub at time $t$ is modeled by a strictly increasing, twice-differentiable function $W$, where $W\left(t\right)$ is measured in degrees Fahrenheit and $t$ is measured in minutes. At time $t = 0$, the temperature of the water is 55$\circ$F. The water is heated for 30 minutes, beginning at time $t =0$. Values of $W\left(t\right)$ at selected times $t$ for the first 20 minutes are given in the table above.
(d) For $20 \leq t \leq 25$, the function $W$ that models the water tempature has first derivative given by $W^{\prime}\left(t\right) = 0.4\sqrt{t}\cos\left(0.06t\right)$. Based on the model, what is the temperature of the water at time $t = 25$?
The answer given by the College board
(d) $W\left(25\right) = 71.0 + \int_{20}^{25} W^{\prime}\left(t\right)\: \textrm{d}t = 71.0 + 2.043155 = 73.043$
What I don't understand is the answer for question d.
the question can be found here and sorry the array didn't work
Best Answer
It must be that you are allowed to use a hand calculator which has numerical integration. When I put the integral into maple, a fairly good antiderivative finder, it didn't find one in closed form. So then I found the numerical approximation to the integral, and that was 2.043144699.. which it seems the answer rounded off to 2.043155, then rounded again in the final answer given, to three decimals.
Or are you just wondering why the formula $$f(b)=f(a)+\int_a^b f'(x) dx$$was used? If that's your question then look up the "fundamental theorem of calculus" which says exactly the above, in one of its forms.