Question

(1,1)(2,1)(2,3)
mirror around y-axis & revolve
solve with Shell and washer
please and thank you

Answer 1

revolve around what? And why reflect it as well?

I will say that the line from (1,1) to (2,3) is
y = 2x-1

You can use that when doing the volume of a shell, which is v = 2πrh dx

Answer 2

revolve around y

Answer 3

well, you are still vague. I assume you mean to revolve the triangle around the y-axis.

So, revolve the small triangle. The volume is

v = ∫[1,2] 2πrh dx
where r = x and h = y-1 = 2x-2
v = ∫[1,2] 2πx(2x-2) dx = 10π/3

to verify, do it again with discs of thickness dy, where

v = ∫[1,3] π(R^2-r^2) dy
where R=2 and r=x=(y+1)/2
v = ∫[1,3] π(2^2-((y+1)/2)^2) dy = 10π/3