The number of cakes brought each year in Illinois from 1990 to 2015 can be approximated by w(t)=412e^(0.071t) (0≥t≤25) with t being the time in years since the start of 1990. How exactly do I find ∫ (with 25 as upper limit and 5 as lower) w(t) dt.

Question

The number of cakes brought each year in Illinois from 1990 to 2015 can be approximated by w(t)=412e^(0.071t) (0≥t≤25) with t being the time in years since the start of 1990.  How exactly do I find ∫ (with 25 as upper limit and 5 as lower) w(t) dt.

oobleck · Answer

since ∫e^(at) = 1/a e^(at), you have ∫[5,25] 412e^(0.071t) dt = 412/0.071 e^(0.071t) [5,25] = 5802.82(e^(.071*25) - e^(.071*5)) = 25962.40