00:01
In this problem, we're given two functions that represent the concentration of a drug when administered intravenously, ci, and then administered orally, co.
00:11
And we're asked to calculate the bioavailability.
00:13
Example 6 of this section poses a very similar problem, and we're told that when calculating bioavailability for each method of administration, it's important to calculate the area under the curve.
00:26
Therefore, we need to take the integral of the function of ci and co, and those bounds are going to go from zero to infinity.
00:36
And then we need to divide the amount for oral over the amount of intravenously, which will give us the final answer of bioavailability.
00:45
So let's start by integrating the ci function.
00:48
So step one, we'll just do intravenously, so we know what we're calculating.
00:58
So we have to take the integral of ci from zero to infinity.
01:02
Already we see that 250 is a constant so we can pull the outside of the integral.
01:07
And we have e to negative 0 .08.
01:12
So now, before we start integrating, we also realize that we have to apply limits because we have infinity.
01:18
So we can do 250 times the limit as a approaches infinity of 0 to a, e to the negative 0 .0 and now we can take the integral so first let's just write the rest of the expression which will give us negative e to negative 0 .08t over 0 .08 and that will be evaluated from 0 to a.
01:45
Now let's plug in those bounds of a and 0.
01:49
We've got 250 times the limit as a approaches infinity.
01:55
So when plugging in a, remember e has a negative power, so we can write that under the denominator so negative 1 over e to the 0 .08 a times 0 .08 minus that since we have a negative sign here will be plus 1 over and then since we'll have e to the 0 that will just be 1 so we'll have 0 .08 so now we need to evaluate the limit of each of the terms the second term is really easy because it's a constant the limit we'll just remain itself.
02:31
So that will be 1 over 0 .08.
02:37
And this can be rewritten actually as 100 over 8, or then you can simplify that to be 25 over 2, 12 .5, which is just easier to write.
02:51
And then for the first term, the denominator will go to infinity as a approach is infinity, meaning that this whole fraction will get infinitely small of finally approaching zero as its limit.
03:05
So now we can write the expression as 250 times, well, our first term is zero, so we can ignore that.
03:14
12 .5, which is the limit of the second term.
03:18
And then in multiplying that, we get 3 ,125 when you plug that into your calculator.
03:23
Okay, so this is our intravenous value, so i'll just put intra, now, what we need to do is calculate the oral administration value.
03:33
So, similarly, now we need to integrate our function from zero to infinity...