How do you find the upper bound of an integral

Question

12000-inte(0 to A) r(t)dt= 9000

 

r(t)= 600t/t+3 for 0 less than or equal to t less than or equal 5

r(t)= t>5

(Piecewise)

 

Answer 1

It would be better if you use parhentheses to make the expressions more clear. For example r(t)=600t/t+3. I assume that is should be r(t)=600t/(t+3).

 

12000-∫₀^Ar(t)dt=9000,  (1)

r(t)= {600t/(t+3), 0≤t≤5,  

         {t, t>5

∫₀^Ar(t)dt=∫₀⁵r(t)dt+∫₅^Ar(t)dt=600∫₀⁵t/(t+3)dt+∫₅^Atdt=

=600(∫₀⁵(1-3/(t+3)dt+A²/2-25/2

=600(5-3ln(8/3))-25/2+A²/2.  Plug in this into (1) and solve for A.