# Surface area of a dome-ish roof

• joebobjoe
In summary, the homework statement is to determine the surface area of the roof of a structure if it is formed by rotating the parabola about the axis. The Attempt at a Solution states that the surface area is 1072.33 m^2. However, according to the website http://www.slideshare.net/mrsbeth63/engineering-mechanics-statics-rc-hibbeler-12th-edition-complete-solutions-ch-9" , the answer is 1365 m^2.

## Homework Statement

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

## Homework Equations

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

## 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?

That's what I get.

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.

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joebobjoe said:
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.

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).

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Dick said:
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).
Why not? How does adding up the circumferences not equal the surface area of the dome.

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

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.

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

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)

Dick said:
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.
Okay thanks.

Calculus is stupid.

## What is the formula for calculating the surface area of a dome-ish roof?

The formula for calculating the surface area of a dome-ish roof is 2πr^2 + 2πrh, where r is the radius of the dome and h is the height of the dome.

## Can the surface area of a dome-ish roof be calculated using the same formula as a regular dome?

Yes, the surface area of a dome-ish roof can be calculated using the same formula as a regular dome. The only difference is that the radius and height used in the formula may be different due to the varying shape of the dome-ish roof.

## What units should be used when calculating the surface area of a dome-ish roof?

The units used for calculating the surface area of a dome-ish roof will depend on the units used for the radius and height of the dome. If the radius is given in feet and the height is given in inches, the final answer will be in square feet.

## What factors can affect the surface area of a dome-ish roof?

The surface area of a dome-ish roof can be affected by the radius and height of the dome, as well as any additional features such as skylights or windows. Changes in the shape or design of the roof can also impact the surface area.

## How is the surface area of a dome-ish roof used in construction?

The surface area of a dome-ish roof is an important factor in construction as it helps determine the amount of materials needed for the roof. It is also used in estimating the cost of construction and can impact the structural integrity of the roof.