# Homework Help: Surface area of a dome-ish roof

1. Nov 3, 2011

### joebobjoe

1. The problem statement, all variables and given/known data

Determine the surface area of the roof of the structure if it is formed by rotating the parabola about the axis.

2. Relevant equations

$SA=\int _0^{16}{2\pi\left ( 4 \sqrt{16-y} \right ) dy}$ (???)

3. The attempt at a solution

$SA=\left [ -\frac{16}{3}\pi\left ( 16-y \right )^{\frac{3}{2}}\right ]_{0}^{16}$
$SA=1072.33$

So, 1072.33 m^2?

2. Nov 3, 2011

### LawrenceC

That's what I get.

3. Nov 3, 2011

### joebobjoe

Well according to page 78 of http://www.slideshare.net/mrsbeth63/engineering-mechanics-statics-rc-hibbeler-12th-edition-complete-solutions-ch-9" [Broken], the answer is 1365 m^2. I kind of understand how they did it, I just want to know why my way doesn't work.

Last edited by a moderator: May 5, 2017
4. Nov 4, 2011

### Dick

Because that's not a formula for surface area. You are integrating 2*pi*f(y). You want to integrate 2*pi*f(y)*sqrt(1+f'(y)^2).

Last edited by a moderator: May 5, 2017
5. Nov 4, 2011

### joebobjoe

Why not? How does adding up the circumferences not equal the surface area of the dome.

6. Nov 4, 2011

### Dick

You are adding up infinitesimal surface areas. If you just use the circumference then you are assuming a cylinder is a good approximation to the cross sectional surface area. It's not. Try applying that to a cone. You'll get the wrong answer.

Last edited: Nov 4, 2011
7. Nov 4, 2011

### ironman1478

surface area is only the outer shell of a solid, which is the SA. what you are doing is finding the area under the curve and multiplying it by 2pi

SA is the arc length (you can think of it as circumference) rotated around 2pi for this problem and the formula for arc length is √(1+[y']^2)

8. Nov 4, 2011

### joebobjoe

Okay thanks.

Calculus is stupid.