3. (10 points) Find the volume of the solid obtained by rotating the region bounded by the line $x=0$ and the portion of the curve $y=x^{\frac{2}{3}}$ that lies in the interval $[0,8]$ about the $y$-axis.

Step 1 ：We are given the curve \(y=x^{\frac{2}{3}}\) and we are asked to find the volume of the solid obtained by rotating the region bounded by the line \(x=0\) and the portion of the curve that lies in the interval \([0,8]\) about the y-axis.

Step 2 ：The volume of the solid obtained by rotating a curve about the y-axis can be found using the formula for the volume of a solid of revolution, which is given by the integral from a to b of pi times the square of the function, with respect to x.

Step 3 ：In this case, the function is \(y=x^{\frac{2}{3}}\), the interval is \([0,8]\), and we are rotating about the y-axis. Therefore, we need to integrate from 0 to 8 of pi times the square of \(x^{\frac{2}{3}}\) with respect to x.

Step 4 ：Performing the integration, we find that the volume of the solid is approximately \(54.8571428571428\pi\) cubic units.

Step 5 ：Final Answer: The volume of the solid is \(\boxed{54.8571428571428\pi}\) cubic units.