Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. x= 1/√(x) , 1<=y<=4 about the y-axis
2
I assume you mean y = 1/√x and x=0
Not sure what the trailing "2" indicates
Anyway, going with the above, you can use discs of thickness dy, so
v = ∫[1,4] πr^2 dy
where r = x = 1/y^2
v = ∫[1,4] π/y^4 dy = 21π/64
Or, you can use shells of thickness dx. Since the boundary includes a straight line segment at the top of length 1/16, the volume will include a small thin cylinder, and then shells of decreasing height.
v = π(1/16)^2(3) + ∫[1,4] 2πrh dx
where r = x and h = y-1
v = 3π/256 + ∫[1/16,1] 2πx(1/√x - 1) dx = 21π/64