Volumes by Slicing (1 Viewer)

[qqmore]

How do you do this problem without cylindrical shells:

The region bounded between the curves y = x and y = 3x - x^2 is rotated about the y - axis. Find the volume generated by taking slices perpendicular to the axis of rotation (y-axis).

Answer is : 8pie/3 units^3.

 

[ngogiathuan]

you have to consider 2 parts of this volume separately, from y=2 to y=0, and the other one from y= 9/4 to y=2, then you find the volume of each slice, and sum up normally. I cant think of any better way
The only hard bit of this volume is when u have to find x in term of y, where y= 3x-x^2. by completing the square
y= 9/4 - [(x-3/2) ^2],
- sqrt (9/4-y) = x - 3/2, taking - sign since when x=0, y =0
so 3/2 - sqrt (9/4-y) = x

Hope this helps