Calculus of a Single Variable, Chapter 7, 7.4, Section 7.4, Problem 41

To calculate the surface area generated by curve y=f(x) revolving about x-axis between a and b, we use the following formula
S_x=2pi int_a^b y sqrt(1+y'^2)dx
Let us therefore first find the derivative y'.
y'=1/(2sqrt(4-x^2))cdot(-2x)=-x/sqrt(4-x^2)
y'^2=x^2/(4-x^2)
We can now calculate the surface.
S_x=2pi int_-1^1sqrt(4-x^2)sqrt(1+x^2/(4-x^2))dx=
2pi int_-1^1sqrt(4-x^2)sqrt(4-x^2+x^2)/sqrt(4-x^2)dx=
2pi int_-1^1 2dx=4pi x|_-1^1=4pi(1+1)=8pi
The area of surface generated by revolving the given curve about x-axis between -1 and 1 is 8pi.
Graphs of the curve and the surface generated by curve's revolution can be seen in the images below.