fundamental theorem of calculus

Question

Around the beginning of the 1800’s, the population of the U.S. was growing at a rate of about 1.34^t  million people per decade, with t being measured in decades from 1810.

If the population P(t) was 7.4 million people in 1810, estimate the population in 1820 (one decade later)

P(1)=

Answer 1

P'(t) = 1.34^t

P(1) = P(0) + P(1) - P(0) = 7.4 + ∫_{(0 to 1)} P'(t)dt

= 7.4 + ∫_{(0 to 1)} (1.34)^t dt

= 7.4 + (1 / ln(1.34)) [ (1.34)^t _{(from 0 to 1)} ] = 7.4 + (1 / ln(1.34)) [1.34 - 1] ≈ 8.6 million

Answer 2

7.4(1.34) = ? (in millions)