Shell method

Question

1. The integral 2 π∫(integral 0 to 1, i dont know how to write it in computer) (1-y) y^2dy represents the volume of a solid of revolution. Identify a) the plane region that is revolved and b) the axis of revolution.

 

2. Use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graph of the equation x=y^2, x=y about the line y=-1

 

 

Answer 1

1. The region bounded by the y-axis, the horizontal line y = 1 and            the graph of x = y² is revolved about the horizontal line y = 1.            The height of a typical shell is y², the radius of the shell is 1-y,            and the thickness of the shell is dy.  So, the volume of the shell          is:

 

        2π(radius)(height)(thickness) = 2π(1-y)(y²)dy.  Integrate               this from y = 0 to y = 1 to obtain the volume of the solid of                 revolution.

 

2. Using the washer method, the volume of a typical washer is               π(1 + √x)²dx - π(1 + x)² dx.  Integrate from x = 0 to x = 1               to get the volume of the solid.