Can anyone help me with this? I know the y axis one but not at x axis

Question

Sketch the graphs of x = (y - 2)^(1⁄2)

, y = x + 4, and the y-axis. Shade the region bounded by the curves. Find

the volume of the solid that is formed by revolving the region shaded about the x-axis.

Answer 1

x = (y-2)^.5 is a vertical parabola with vertex (0,2) opening upward

y-2 = x^2

y = x^2 +2

y intercept = the vertex = y=2

y = x+4 is a straight line with y intercept =4, x intercept = -4

set them equal to find where they intersect

x=y-4 = (y-2)^.5

y^2 -8y +16 = y-2

y^2 -9y +18 = 0

(y-6)(y-3) = 0

y = x+4 = 3,6

x = y-4 = -1,2

intersection points are (-1,3) and (2,6)

to find the volume use a double integral

limits of integration are from x=-1 to 2, horizontally

and from y=3 to 6 vertically

use shell or disc method

disc is generally easier to use when rotated around the x axis