Centroid and Area Problem

[Turan jae]

Hi, I am in Calculus II in college and I have been stuck on this problem for a long time. I would really appreciate if someone could break down how to solve this.

R is the region bounded by y = x +x, y = 0, and x = 2.
S is the solid obtained by rotating R about x = 5.
T is the solid obtained by rotating R about y = 0.

• Provide integrals for the x an y values of the centroid of R.
• Provide an integral for the volume of S. (know how to do this one)
• Provide an integral for the surface area of the boundary surface of T.

 

[HallsofIvy]

"y= x+ x" seems very peculiar as it would normally be written "y= 2x". I am inclined to think that second x was supposed to be a specific number. In any case, the centroid of a plane region, R, is the point \(\displaystyle \left(\overline{x}, \overline{y}\right)\) where \(\displaystyle \overline{x}= \int_R x dV\) and \(\displaystyle \overline{y}= \int_R y dV\).