Answer
$$\frac{{45}}{4}$$
Work Step by Step
$$\eqalign{
& \int_1^8 {\root 3 \of y } dy \cr
& {\text{use property of radicals }}\root n \of {{a^m}} = {a^{m/n}} \cr
& = \int_1^8 {{y^{1/3}}} dy \cr
& {\text{integrate using the power rule for integration}} \cr
& = \left. {\left( {\frac{{{y^{4/3}}}}{{4/3}}} \right)} \right|_1^8 \cr
& = \frac{3}{4}\left. {\left( {{y^{4/3}}} \right)} \right|_1^8 \cr
& {\text{using the fundamental theorem}} \cr
& = \frac{3}{4}\left( {{{\left( 8 \right)}^{4/3}} - {{\left( 1 \right)}^{4/3}}} \right) \cr
& {\text{simplify}} \cr
& = \frac{3}{4}\left( {16 - 1} \right) \cr
& = \frac{{45}}{4} \cr} $$
