help again with word problem?

Question

During a surge in the demand for electricity, the rate, r, at which energy is used can be approximated by the following where t is the time in hours and p is a positive constant.

r = te^(-pt)

(a) Find the total energy (E) used in the first (T) hours. Give your answer as a function of p.

(b) What value does E approach as T approaches infinity?


please, help me. can you walk me through it so i understand it? sorry, i posted this question earlier, but it copied wrong.

Answer 1

a)
your given
r = dE/dt = te^(-pt) so
E(t) = ∫te^(-pt) dt = (1/p^2)(pt + 1)e^-pt
using integration by parts
then evaluating E(t) from 0 to T gives
E(T) - E(0)
(1/p^2)(pT + 1)e^-pT - (1/p^2)
so
E(p) = (T/p)e^-pT
is E as a function of p
[[[ note: E(0) = (1/p^2) since e^0 = 1 ,which the above answerer forgot ]]]

then
b)
lim (T/p)e^-pT = lim T/(pe^pT) = 0 as T---->∞

.

Answer 2

∫te^(-pt) = (1/p^2)(pt + 1)e^-pt
(a) E = (1/p^2)(pT + 1)e^-pT
(b)
lim(pT + 1)/(p^2e^pT) =
T→∞

lim[p/(p^3e^pT)] = 1/p^2
T→∞