y=x;y=square root (x); about y=1. Find the volume using washer/disk method.

Question

I got pi/6. Not sure about the answer. Thanks.

Answer 1

The region bounded by y = x and y = √x lies in the first quadrant with x in the interval [0,1].  

 

Take a thin vertical cross section of the region with width dx.  Rotating the cross section about the line y = 1(a horizontal line) generates a washer with outer radius 1-x and inner radius 1-√x.

 

Note:  With the disk/washer method, we always take cross sections               perpendicular to the axis of rotation.  

 

Volume of typical washer = π(1-x)²dx - π(1-√x)²dx

 

                                    = π[1-2x+x²-1+2√x-x]dx

 

                                    = π[x²+2√x-3x]dx

 

Volume of solid = π∫_{from 0 to 1}[x²+2√x-3x]dx

 

                       = π[(1/3)x³+(4/3)x^3/2-(3/2)x²]_{from 0 to 1}

 

                       = π[(1/3)+(4/3)-(3/2)] = π/6

 

Ab:  I guess that we are both right (or both wrong).