Answer
$ \int_{0}^{30} u(t) dt=-C_{0} [e^{\frac{-30r}{V}}-1]$
Work Step by Step
Dialysis treatment removes urea and other waste products from a patient’s blood by diverting some of the blood flow externally through a machine called a dialyzer.
The rate at which urea is removed from the blood (in mg/min) is often well described by the equation
$u(t)=\frac{r}{V}C_{0} e^{\frac{-rt}{V}}$
Where, $r $ is the rate of flow of blood through the dialyzer (in mL/min), $V$ is the volume of the patient’s blood (in mL), and C_{0} is the amount of urea in the blood (in mg) at time .
Evaluate the value of the integral $ \int_{0}^{30} u(t) dt$
Thus,
$ \int_{0}^{30} u(t) dt=\int_{0}^{30} \frac{r}{V}C_{0} e^{\frac{-rt}{V}}dt$
$=\frac{r}{V}C_{0} (-\frac{V}{r})[e^{\frac{-rt}{V}}]_{0}^{30}$
Hence, $ \int_{0}^{30} u(t) dt=-C_{0} [e^{\frac{-30r}{V}}-1]$
