$[4$ marks$]$ Express the following limits as a definite integral on the interval $0,1.$

$($a$)\underset{n}{lim}\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{}}}\sqrt[\phantom{2}]{4+\frac{5i}{n}}.$

$($b$)\underset{n}{lim}{\displaystyle \underset{j=1}{\overset{n}{}}}\frac{j^7}{n^8}.$