Find the volume of the solid of revolution generated by revolving the region bounded by y = 6, y = 0, x = 0, and x = 4 about: the x–axis 452.389 y–axis 301.593

Question

AP Calculus question ,need help!!

Answer 1

The region is a rectangle in the first quadrant with one side lying on the x-axis and another side along the y-axis. 

 

Rotate about the x-axis:  Use the disk method.  A typical slice perpendicular to the x-axis has height 6 and thickness Δx.  Rotating the slice about the x-axis yields a disk of radius 6 and thickness Δx.

 

                                     Volume of disk = π(6)²Δx = 36πΔx

 

                                     Volume of solid = 36π∫_{(from 0 to 4)}dx = (36πx)_{(from 0 to 4)} = 144π ≈ 452.389

 

            Note:  We can also do this problem without resorting to the methods of Calculus.  The solid is a cylinder     with radius 6 and height 4.  So, Volume of cylinder = π(radius)²(height) = 144π.

 

Rotate about the y-axis:  Use the disk method.  Rotating a cross section perpendicular to the y-axis yields a disk    

                                     with radius 4 and thickness ΔΔ.

 

                                     Volume of disk = π(4)²Δy = 16πΔy

 

                                     Volume of solid = 16π∫_{(from 0 to 6)} dy = (16πy)_{(from 0 to 6)} = 96π ≈  301.593

 

          The volume can also be found by using the formula for the volume of a cylinder.