007B Sample Midterm 3, Problem 3 Detailed Solution

For a fish that starts life with a length of 1cm and has a maximum length of 30cm, the von Bertalanffy growth model predicts that the growth rate is  $29e^{-t}$  cm/year where  $t$  is the age of the fish. What is the average length of the fish over its first five years?

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Background Information:
The average value of a function  $f(x)$  on an interval  $[a,b]$  is given by
$f_{\text{avg}}={\frac {1}{b-a}}\int _{a}^{b}f(x)~dx.$

Solution:

Step 1:
Let  $f(x)=\sin(x).$
Each interval has length  ${\frac {\pi }{4}}.$
Therefore, the right-endpoint Riemann sum of  $f(x)$  on the interval  $[0,\pi ]$  is
${\frac {\pi }{4}}{\bigg (}f{\bigg (}{\frac {\pi }{4}}{\bigg )}+f{\bigg (}{\frac {\pi }{2}}{\bigg )}+f{\bigg (}{\frac {3\pi }{4}}{\bigg )}+f(\pi ){\bigg )}.$

Step 2:
Thus, the right-endpoint Riemann sum is
${\begin{array}{rcl}\displaystyle {{\frac {\pi }{4}}{\bigg (}\sin {\bigg (}{\frac {\pi }{4}}{\bigg )}+\sin {\bigg (}{\frac {\pi }{2}}{\bigg )}+\sin {\bigg (}{\frac {3\pi }{4}}{\bigg )}+\sin(\pi ){\bigg )}}&=&\displaystyle {{\frac {\pi }{4}}{\bigg (}{\frac {\sqrt {2}}{2}}+1+{\frac {\sqrt {2}}{2}}+0{\bigg )}}\\&&\\&=&\displaystyle {{\frac {\pi }{4}}({\sqrt {2}}+1).}\\\end{array}}$

Final Answer:
${\frac {\pi }{4}}({\sqrt {2}}+1)$

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