Question

Find the exact area of the surface obtained by rotating the curve about the x-axis.

9x = y2 + 36, 4 ≤ x ≤ 8

I keep on getting 4pi (3^(3/2) +1)

so I tried to put the answer in that way and also 24.78pi
But it is wrong

Answer 1

9x = y^2 + 36

I assume you just mean the part above the x-axis, since the curve is symmetric about that axis. So, using discs, the volume is

v = ∫[4,8] πr^2 dx
where r^2 = y^2 = 9x-36
v = π∫[4,8] 9x-36 dx
= π(9/2 x^2 - 36x) [4,8]
= 72π

Using shells, we have

v = ∫[0,6] 2πrh dy
where r = y and h = (8-x) = 8-(y^2+36)/9)
v = 2π∫[0,6] y(8-(y^2+36)/9) dy
2π(2y^2 - y^4/36) [0,6]
= 72π

Answer 2

I am just suppose to find the surface area. We are studying arc length

Answer 3

Oops. Oh well. The observation about symmetry still holds. The surface area is just the sum of circumferences of many circles of radius y. So,

dS = 2πy ds
S = 2π∫[4,8] y√(1+y'^2) dx

9x = y^2+36, so y = 3√(x-4)
18 = 2yy'
y' = 9/y = 9/√(9x-36)
so, y'^2 = 81/(9x-36) = 9/(x-4)

S = 2π∫[4,8] (3√(x-4))√((x+5)/(x-4)) dx
= 6π∫[4,8] √(x+5) dx
= 4π(x+5)^(3/2) [4,8]
= 79.49 π

Better double-check my math.

Answer 4

I redid the problem using polar coordinates, where I set the (0,0,0) at the base of the paraboloid section, so that

z = 4 - (x^2+y^2)/9

That means that we have only to evaluate

S = 1/9 ∫[0,2π]∫[0,6] r√(81+4r^2) dr dθ
and we get a nice tidy 49π

Not sure where I went wrong using x = g(y) and rectangular coordinates. You can see a discussion of the coordinate change at

http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/surface/surface.html