# An experimental drug lowers a patient's blood serum cholesterol at the rate of t(100 - t^2)^{1/2}...

## Question:

An experimental drug lowers a patient's blood serum cholesterol at the rate of {eq}t(100 - t^2)^{1/2} {/eq} units per day, where t is the number of days since the drug was administered {eq}(0 \leq t \leq 10) {/eq}. Find the total change during the first 6 days.

## Integrals:

Whenever we see a rate of change, what we are really looking at is a derivative. So if we want to find how much something has changed over a specific interval and we know the rate of change, then we can use the fact that derivatives and integrals are inverse processes of each other (i.e. we use the fundamental theorem of calculus) to find that total change using integration.

We have the rate of change, so to get the total change, we just integrate the rate over the interval. So the total change during the first 6 days is

{eq}\begin{align*} \int_0^6 t (100-t^2)^{1/2}\ dt &= \left [ -\frac13(100-t^2)^{3/2} \right ]_0^6 \\ &= \frac13 \left[ - (100-6^2)^{3/2} + (100-0^2)^{3/2} \right] \\ &= \frac{488}{3} \\ &\approx 162.6667 \end{align*} {/eq}