by Guest » Sun Dec 19, 2021 12:50 pm
It would be nice if they told us what "[tex]V_1[/tex]" and "[tex]V_2[/tex]" stood for!
I am going to guess that [tex]V_1[/tex] is the volume of the entire region from the x-axis up to the graph of y= f(x) between x= 1 and x= 4, rotated around the x-axis, [tex]--\pi \int_1^4 f^2(x)dx[/tex], and that [tex]V_2[/tex] is the volume of the region from the x-axis up to y= 1 rotated around the x-axis. That last is just a cylinder of radius 1 and length 3 so its volume is [tex]3\pi[/tex].
That is, [tex]\pi \int_1^4 f^2(x)dx- 3\pi= 4\pi[/tex] so you know that [tex]\pi \int_1^4 f^2(x)dx= 7\pi[/tex] and you are asked to find [tex]\int_1^4 f(x)dx[/tex] minus the area below the shaded region which is 4.
(It occurs to me that we could use the "theorem of pappus" which says that the volume of a region, rotated around the x-axis, is the area of the region times the distance from the x-axis to the centroid of the area, but that requires finding the centoid!)