How do you find the volume of the solid generated by revolving the plane region bounded by the graphs of #x^2 = y - 2# and 2y - x -2 = 0 about the line y =3 with x =0, x=1?

Question

How do you find the volume of the solid generated by revolving the plane region bounded by the graphs of #x^2  = y - 2# and 2y - x -2 = 0 about the line y =3  with x =0, x=1?

Isaiah Allen · Accepted Answer

I have solved this way:

Emily Ammerman · Answer

undefined To find the volume of the solid generated by revolving the given plane region about the line y = 3, we first need to find the points of intersection of the two curves and then use the method of cylindrical shells.

1. Find the points of intersection of the curves x^2 = y - 2 and 2y - x - 2 = 0.
   Solve the system of equations to find the intersection points.

2. Once you have the intersection points, integrate the expression for the circumference of the cylindrical shells times the height of each shell from x = 0 to x = 1.

The expression for the circumference of the cylindrical shell is 2π(radius), where the radius is the distance from the axis of rotation (y = 3) to the curve. The height of each shell is the difference in y-values between the two curves.

3. Set up the integral:

∫[0,1] 2π(radius) * (height) dx

4. Evaluate the integral to find the volume of the solid.