Brief Calculus.

Question

The number of patients admitted to a large inner-city hospital is changing by 𝑝′(ℎ)=0.02ℎ³ −0.6ℎ² +0.4h+10

patients per hour h hours after midnight.

a) Find the value and interpret the meaning of

∫⁴₀ 𝑝′(ℎ) 𝑑ℎ

b) Find the value and interpret the meaning of

∫¹²₀ 𝑝′(ℎ) 𝑑ℎ

Answer 1

p(h) is the antiderivative of p'(h), so p(h) = 0.005h^4 - 0.2h^3 + 0.2h^2 + 10h + C

for a), the integral of p'(h) from 0 to 4 is p(4) - p(0), which = 31.68. That's (roughly) the number of total patients admitted by 4 AM (i.e., four hours after midnight).

for b) the integral is p(12) - p(0) = -93.12. That means that by noon the next day (12 hrs after midnight), the hospital has 93 fewer patients than were there at midnight.