The rate of change of the mass of a substance in a chemical reaction, M′(t)M'(t)M′(t) in grams per hour, is modeled by the function:
M′(t)=18t3t+4,0≤t≤4 M'(t) = \frac{18t}{\sqrt{3t+4}}, \quad 0 \le t \le 4 M′(t)=3t+418t,0≤t≤4Using the substitution u=3t+4u = \sqrt{3t+4}u=3t+4, find the exact change in mass, in grams, between t=0t = 0t=0 and t=4t = 4t=4.
In a separate experiment, the pressure within a vessel is found to vary according to the function:
f(p)=6p2−11(p−1)(3p+2) f(p) = \frac{6p^2 - 11}{(p - 1)(3p + 2)} f(p)=(p−1)(3p+2)6p2−11Find ∫f(p) dp\int f(p) \, dp∫f(p)dp.