How do you find the volume of the region enclosed by the curves #y=x#, #y=-x#, and #x=1# rotated about the y axis?

2 Answers
Oct 10, 2015

If rotating about the y-axis, the shell method is your best bet to a quick solution.

Explanation:

enter image source here

#Volume=2piint_0^1x[x-(-x)]dx=2piint_0^1(2x^2)dx#

#=4/3pi#

hope that helped

Oct 10, 2015

Look at the big picture then break it apart into extrusions.

Explanation:

https://www.desmos.com/calculator/zem7gewjxm

I see it like a (Cylinder-2Cone) instead of a funky shape. So if we find the volume of the cylinder we get #2pi# because #pir^2h#.

Now we have 2 Cones which are #pir^2h/3# which turns out to be #pi/3# because the height is now one. Now we take #2pi#-#2(pi/3)# or a total #V=(4pi)/3#