Question

Fluid leaks out of a bottle at the rate of ln(1+t)+cos(t+ sqrt(t)) milliliters per minute, where t is measured in minutes. if the bottle contains 375 ml of fluid at time t=0 min, how many ml of fluid are in the bottle at time t=60?

Answer 1

You want \(F(60)\)
Given: \[
F'(t) = \ln(1+t)+\cos(t+ \sqrt{t})\\
F(0) = 375
\]
Just use the fundamental theorem of calculus: \[
\int_0^{60}\ln(1+t)+\cos(t+ \sqrt{t})dt = F(60) - 375
\]

Answer 2

Basically that formula comes from: \[
\int_a^b F'(t)dt = F(b) - F(a)
\]