Calculus 2 Volumes and Volumes by Cyl. Shells HW problem.

Question

Problem:

Graph the following enclosed region:

y=-x^2+11x , y=0, x=0

State the method that be used to find the volume of the solid created by rotating the region about the following axis of rotation:

a) y-axis(line x=0) b) line y=2

For each part above draw the region with a representation rectangle (graphing needed) and label the endpoints of the interval spanned and any radius and/or height needed in finding the volume

a) b)

Answer 1

There appear to be errors in the statement of this problem: the line x = 0 does not help enclose anything and is redundant.

I assume the revolved region is the lobe between the vertex of the parabola and the x axis (line y = 0).

(a) Use Cylindrical shells of radius x, thickness dx, and height y(x) = -x^2 + 11x. Integrate for x = 0 to x = 11.

The integral will be 2 pi x y(x) dx

(b) Revolving around y = 2 will probably slice the volume into vertical disks of thickness dx, and radius r(x) = y(x) - 2.

However the lower part of the revolved region (between the lines y=2 and y=0) will sweep into the volume of the upper part of the revolved region so it is not clear how that should be treated. Again it appears that the problem is not stated well - probably incompetent instructor.