Find the volume of the solid

Question

Find the volume of the solid generated by revolving the region bounded by the graphs of y=x y=sqrtx about the line y=1

Answer 1

Using the washer method, a typical vertical cross section of the region has outer radius 1-x and inner radius 1-√x.  The curves intersect when x = 0 and x = 1.  

 

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

 

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

 

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

 

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

 

                        = 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