Surface Area Equality: The Simplest Explanation

• calculushelp
In summary, the surface area of a function in 3D, for which the surface z(x,y) is given, is calculated by taking the integral of \sqrt{\left(\frac{\partial z}{\partial x} \right)^2 + \left(\frac{\partial z}{\partial y}\right)^2 + 1} \ \ dA over the region R. This formula resembles the one for calculating arc length and is used for surfaces of rotation. The given equations do not make sense as they do not include an integral sign.
calculushelp
ok why does

surface area = ( y) ( 1+ f ' (x)^(2) )^(1/2)

equal

surface area = (x ) (1+ f ' (x)^(2) )^(1/2)

the simplest explanation please.

Where did you get those formulae from? And surface area of what? You're supposed to do a surface integral in 3D space (the details of which depend on the parametrisation of the surface area)to get the surface area, but I don't see any integral sign. The formulae you gave resembles the one given for arc length calculation.

For a surface area of a function in 3D, for which the surface z(x,y) is given, the surface area of the portion that projects down onto a region R is given by $$\int \int_R \sqrt{\left(\frac{\partial z}{\partial x} \right)^2 + \left(\frac{\partial z}{\partial y}\right)^2 + 1} \ \ dA$$

calculushelp said:
ok why does

surface area = ( y) ( 1+ f ' (x)^(2) )^(1/2)

equal

surface area = (x ) (1+ f ' (x)^(2) )^(1/2)

the simplest explanation please.

There can be no explanation- what you have written makes no sense at all! Are there supposed to be integrals in there?
Are you talking about the area of surfaces of rotation?

yes! in rotation! sorry i didnt know how to put an integral sign.

What is surface area equality?

Surface area equality refers to the concept that two objects can have the same surface area, but different shapes. This means that the total area of the surface of both objects is equal, even though the objects may look different.

How is surface area equality calculated?

Surface area equality is calculated by finding the total area of the surface of each object and comparing them. This can be done by using mathematical equations or formulas specific to the shape of the object.

Why is surface area equality important?

Surface area equality is important because it helps us understand that even though two objects may look different, they can still have the same surface area. This concept is used in many fields of science, including physics, chemistry, and biology.

Can surface area equality be applied to real-life situations?

Yes, surface area equality can be applied to real-life situations. For example, two different shaped swimming pools can have the same surface area, but different depths.

How does surface area equality relate to the conservation of matter?

Surface area equality relates to the conservation of matter because it shows that even though the shapes of two objects may change, the total amount of matter remains the same. This is because the surface area, and therefore the number of particles, remains constant.

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