Integration and population dynamics?

[ineedhelpnow]

can someone explain this problem step by step (not a homework problem, just an example i found and i want to see how its done).

a hot wet summer is causing a mosquito population explosion in a lake resort area. the number of mosquitoes is increasing at an estimated rate of 2200+10e^(0.8t) per week (t is in weeks). by how much does the mosquito population increase between the fifth and ninth weeks of summer.

 

[MarkFL]

We are given:

$$\d{P}{t}$$

and asked to find:

$$N=P(9)-P(5)$$

Can you see how to use the anti-derivative form of the FTOC here?

 

[ineedhelpnow]

no do you mind explaining?

 

[Jameson]

If you are given $dP$ what operation can you use to find $P$? If you want to find $P(a)-P(b)$ what operation would you use from $dP$? Something should come to mind.

 

[ineedhelpnow]

this is probably wrong but do i just integrate 2200+10e^0.8t from 5 to 9?

@Jameson this sounds super weird but I get a happy feeling whenever i see your username and your profile picture. its' so cool man, i swear. :D

 

[MarkFL]

this is probably wrong but do i just integrate 2200+10e^0.8t from 5 to 9?

Let's take a look:

$$P(9)-P(5)=\int_5^9\,dP=\int_5^9\d{P}{t}\,dt$$

Yep, looks good to me. :D

 

[ineedhelpnow]

soooo...24860? does that sound right?

 

[MarkFL]

Yes, since a population is discrete rather than continuous we would use:

$$N=\left\lfloor \int_5^9 2200+10e^{0.8t}\,dt \right\rceil=24860$$

 

[ineedhelpnow]

ok thanks.