Use cylindrical shells to find the volume of the solid obtained by rotating the region bounded by y=x^2 , y=0, and x=7, about the y-axis.

Answer 1

Volume v = 2π∫₀⁷x·x²dx = 2π·x⁴/4₀⁷ = 2401π/2

Answer 2

When you're dealing with Cylindrical shells, you're basically creating a rectangle with the height between the functions (x²-0) and the width of Δx (called dx in an integral) and rotating that around the y-axis (radius = x). That being said, the volume of each of these shells would be that area (x²dx) times the circumference (2πx):

V = 2πx³dx. Then you integrate this between the bounds of 0 and 7 (area between the y=0 and y=x² begins at 0 and x=7 is the set ending). This gives the total volume of ^π/₂(7⁴) cubic units or 2401π/2 cubic units.