Question 556723
First, draw the graph of the function and the rotation. You didn't specify which axis you rotate the region about. I'll assume you rotate about the x-axis.


We want to evaluate the integral


*[tex \LARGE \int_{0}^{2} \pi (x^3)^2 \, dx = \pi \int_{0}^{2} x^6 \, dx]


This is equal to


*[tex \LARGE \frac{\pi x^7}{7}] evaluated at x = 2, minus the value at x = 0, or simply


*[tex \LARGE \frac{128 \pi}{7}].