Shell Method; Washer Method

Question

Find the volume of the solid of revolution formed when the region bounded by the given curves is revolved about the specified axis. REGION:  y=tan x ; y=0 ; & x = pi/4           AXIS: x-axis

Answer 1

 

Hi Jacob,

 

As it is the revolution about x-axis,

 

the volume we need to calculate is ∫^b_a π · (radius)² dx

 

where a = 0, b = π/4, and radius (at x) = tan(x).

 

     ∫^π/4₀ π · tan²(x) dx = π∫^π/4₀ tan²(x) dx

 

                                 = π∫^π/4₀ (sec²(x) - 1) dx

                

                                   ( from the identity, 1 + tan²(x) = sec²(x) )

 

                                 = π · (tan(x) - x)|^π/4₀

 

                                 = π · ((tan(π/4) - π/4) - (tan(0) - 0))

 

                                 = π · (1 - π/4)