Math $1$A

Definite Integrals are Everywhere

$3.$ Using the results of question two, determine what the balance will be in a bank account after one year where,

The initial balance at time $t=0$ is zero.

Interest is compounded continuously with annual rate $r=0\mathrm{.}05($this would be $5\%$ in everyday language$).$

Money is continuously added to the account at rate $f(t)=10000t{e}^{t^2+0\mathrm{.}05t}$

If$,$ instead of continuously adding money, we made a single deposit at time $t=0,$ how much would it need to be to guarantee the same balance after a year?