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 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 \int_a^b r'(t)~dt,} what are we calculating?
We are calculating 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 r(b)-r(a).} This is the total reaction to the
drug from 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 t=a} to 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 t=b.}

Solution:

Step 1:

Step 2:

Step 3:

Final Answer:

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