For each x , 0 ≤ x ≤ 1 {\displaystyle x,0\leq x\leq 1} , the point ( x , f ( x ) ) {\displaystyle (x,f(x))} is rotate to form a circle of radius
r ( x ) = f ( x ) − ( − 1 ) = f ( x ) + 1 = x 2 + 1. {\displaystyle r(x)=f(x)-(-1)=f(x)+1=x^{2}+1.}
The area of this circle is thus π r ( x ) 2 = π ( x 2 + 1 ) 2 = π ( x 4 + 2 x 2 + 1 ) {\displaystyle \pi r(x)^{2}=\pi (x^{2}+1)^{2}=\pi (x^{4}+2x^{2}+1)} . The volume V {\displaystyle V} of the surface formed by putting together all these circles is just the integral of their respective areas:
V = ∫ 0 1 π ( x 4 + 2 x 2 + 1 ) d x = π [ 1 5 x 5 + 2 3 x 3 + x ] 0 1 = π ( 1 5 + 2 3 + 1 ) = 28 15 π . {\displaystyle V=\int _{0}^{1}\pi (x^{4}+2x^{2}+1)\mathrm {d} x=\pi \left[{\frac {1}{5}}x^{5}+{\frac {2}{3}}x^{3}+x\right]_{0}^{1}=\pi \left({\frac {1}{5}}+{\frac {2}{3}}+1\right)={\frac {28}{15}}\pi .}
, the point 
is rotate to form a circle of radius
. The volume 
of the surface formed by putting together all these circles is just the integral of their respective areas:
![{\displaystyle V=\int _{0}^{1}\pi (x^{4}+2x^{2}+1)\mathrm {d} x=\pi \left[{\frac {1}{5}}x^{5}+{\frac {2}{3}}x^{3}+x\right]_{0}^{1}=\pi \left({\frac {1}{5}}+{\frac {2}{3}}+1\right)={\frac {28}{15}}\pi .}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/e5925ad006049c9117c2e5d0e7ac650798bd4038)
.jpg)