The volume of the solid thus formed is [tex]\frac{26}{27} \pi[/tex] .
Let us think of any region that is enclosed by a function g(x) and the x-axis between x=a and x=b. If this region is rotated about the x-axis, the solid of rotation that results can be thought of as consisting of thin cylindrical discs with a width of dx and a radius of g (x). Therefore, by integrating the volume of this disc, the whole volume of the solid will be revealed.
Volume=[tex]\int^{a}_{b}\pi [g(x)]^2dx[/tex]
The given function is g(x)=(1/3)x+2 ranging from x=1 to x=3
Therefore volume of the solid formed is:
[tex]\int^{3}_{1}\pi [\frac{1}{3}x]^2dx\\=\int^{3}_{1}\pi [\frac{1}{9}x^2]dx\\=\pi \int^{3}_{1} [\frac{1}{9}x^2]dx\\\\=\pi [\frac{1}{27}x^3 ]^3_1\\= \frac{26}{27} \pi[/tex]
The volume of the solid thus formed is [tex]\frac{26}{27} \pi[/tex] .
To learn more about volume of solids around a curve:
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