Question

Find the exact area of the surface obtained by rotating the curve about the x-axis.
9x = y^2 + 27, 3 ≤ x ≤ 7

Answer 1

Should I get y in terms of x first?

Answer 2

I would not bother.
notice we can write this as
x= (1/9) y^2 + 3 3 <= x < 7
which is a ( shaped parabola

if you like you could switch all the x and y's:

y = (1/9) x^2 + 3 3 <= y <= 7 (corresponding to x=0 to 6)
rotated about the y-axis

Answer 3

using y = (1/9) x^2 + 3 3 <= y <= 7
I get \[SA = 2\pi \int\limits_{0}^{6}(\frac{ 1 }{ 9 }x^2+3)\sqrt{\frac{ 4 }{ 81 }x^2 +1} dx\]

Answer 4

is this correct?

Answer 5

using the "switched x-y version"