A general question about Surface Area

In summary, this question is about the surface area of a function defined as f(x) rotated around the x-axis. Different integrals wouldn't work to find the surface area because the sum of the delta-x's would add up to 1.
  • #1
kscribble
10
0
This question is about the surface area of a function defined as f(x) rotated around the x-axis

Now I understand how the ACTUAL integral works to find the surface area, but I'm wondering why a different integral wouldn't work...

[tex]\int[/tex] 2[tex]\pi[/tex]*f(x) dx

wouldn't this add up the differential circumferences, so shouldn't this equation work? why do you have to use the Pythagorean theorem integral version to find the surface area?

thanks if you can answer this question :)
 
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  • #2
That doesn't work for the same reason such a technique doesn't work for calculating arc length. Look at the graph of y = x from 0 to 1. We know its length is sqrt(2). But suppose you make a staircase function going up that line and use the sum of the delta-x's as your approximation to the length. Even when you take a finer and finer staircase, the sum of the delta-x's adds up to 1. The horizontal steps even though they get closer and closer to the straight line, don't approach it in length. Your surface area example is the same thing in one higher dimension.
 
  • #3
LCKurtz said:
That doesn't work for the same reason such a technique doesn't work for calculating arc length. Look at the graph of y = x from 0 to 1. We know its length is sqrt(2). But suppose you make a staircase function going up that line and use the sum of the delta-x's as your approximation to the length. Even when you take a finer and finer staircase, the sum of the delta-x's adds up to 1. The horizontal steps even though they get closer and closer to the straight line, don't approach it in length. Your surface area example is the same thing in one higher dimension.

very interesting... i really never thought of it like that. damn, thank you!

you really know your stuff :)
 

Related to A general question about Surface Area

1. What is surface area?

Surface area refers to the total measurement of the exposed outer surface of an object. It is typically measured in square units, such as square inches or square meters.

2. How is surface area calculated?

The formula for calculating surface area varies depending on the shape of the object. For example, the surface area of a cube is calculated by multiplying the length of one side by itself and then multiplying that by six. For more complex shapes, such as a sphere or cylinder, there are specific formulas that take into account the radius and height of the object.

3. Why is surface area important?

Surface area is important because it helps us understand the physical properties and characteristics of objects. It is also an important factor in various real-world applications, such as determining the amount of material needed for construction or calculating the rate of heat transfer.

4. How does surface area relate to volume?

Surface area and volume are closely related, as they both measure different aspects of an object's size. Surface area measures the outer surface of an object, while volume measures the amount of space inside the object. In general, as surface area increases, so does volume.

5. Can surface area be changed?

Yes, surface area can be changed by altering the shape or dimensions of an object. For example, stretching a rubber band will increase its surface area, and folding a piece of paper will decrease its surface area.

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