How do I pick the shell radius for this region rotated about the y-axis?

[Mark.P]

I'm working through a problem about finding the volume of a solid of revolution using cylindrical shells, and I'm stuck on setting up the integral. The region is bounded by y = sqrt(x), y=0, and x=4, rotated about the y-axis. I keep getting confused about whether the radius of a shell is just 'x' or if it's something else when the axis is vertical.

[TimothyW]

Around the y axis you use vertical shells. The shell at position x has radius equal to the distance from the y axis, which is x. The height is the vertical extent of the region, from y = 0 up to y = sqrt(x). So height = sqrt(x). Set up V = ∫ from 0 to 4 of 2 pi (radius)(height) dx = ∫ 0^4 2 pi x sqrt(x) dx.

[Frank11]

I kept thinking the radius was something like 4 minus x or sqrt(x) itself. But with rotation about the y axis the radius is the distance to the axis, so for a strip at x it’s just x.

[RichardPL]

Do I need to worry about the left edge near x = 0 where the region touches the axis? I mean does that change the radius or height in any way?

[RobertUS]

I tried sketching it and it felt off at first, like I was mixing washers and shells. But as long as I keep the strip at x, radius x and height sqrt(x), it should be fine. If you actually carry through the integral you’ll land on a number, but I’m not confident about the exact value off the top of my head.