009B Sample Midterm 3, Problem 4

The rate of reaction to a drug is given by:

$r'(t)=2t^{2}e^{-t}$

where  $t$  is the number of hours since the drug was administered.

Find the total reaction to the drug from  $t=1$  to  $t=6.$

Foundations:
If we calculate  $\int _{a}^{b}r'(t)~dt,$  what are we calculating?
We are calculating  $r(b)-r(a).$  This is the total reaction to the
drug from  $t=a$  to  $t=b.$

Solution:

Step 1:
To calculate the total reaction to the drug from  $t=1$  to  $t=6,$
we need to calculate
$\int _{1}^{6}r'(t)~dt=\int _{1}^{6}2t^{2}e^{-t}~dt.$

Step 2:
We proceed using integration by parts.
Let  $u=2t^{2}$  and  $dv=e^{-t}dt.$
Then,  $du=4t~dt$  and  $v=-e^{-t}.$
Then, we have
$\int _{1}^{6}2t^{2}e^{-t}~dt=\left.-2t^{2}e^{-t}\right|_{1}^{6}+\int _{1}^{6}4te^{-t}~dt.$

Step 3:
Now, we need to use integration by parts again.
Let  $u=4t$  and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dv=e^{-t}dt.}
Then,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle du=4dt}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v=-e^{-t}.}
Thus, we get
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\int_1^62t^2e^{-t}~dt} & = & \displaystyle{\left. (-2t^2e^{-t}-4te^{-t})\right|_1^6+\int_1^6 4e^{-t}}\\ &&\\ & = & \displaystyle{\left. (-2t^2e^{-t}-4te^{-t}-4e^{-t})\right|_1^6}\\ &&\\ & = & \displaystyle{(-2(6)^2e^{-6}-4(6)e^{-6}-4e^{-6})-(-2(1)^2e^{-1}-4(1)e^{-1}-4e^{-1})} \\ &&\\ & = & \displaystyle{\frac{-100+10e^5}{e^6}.} \end{array}}

Final Answer:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{-100+10e^5}{e^6}}

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