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Ascendant0

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For example, say there is a trumped-shaped object whose outer shell is defined by ## y=x^2 ## that spins around the x-axis from x = 0 to x = 3. If I want to find the volume within that object, it would be:

## \int_{0}^{3} \pi x^4 dx ## // I believe this is correct, basically the integral of the area of all "pieces" of circles created by dx from 0 to 3, right?

Since dx gives the location of the outside of that object at any given point, if I wanted to then calculate the surface area of that same object, my initial thoughts were you simply do the same thing, but for the circumference of the circle

*(## 2 \pi r ##) applied*to the equation

*, giving:*

## \int_{0}^{3} 2 \pi x^2 dx ## // I feel since dx gives the location of the outer edge at any point, simply calculate the total of the outer edge's circumference from 0 to 3, and that would be it

Now, since I've already made this lengthy, in short, I have seen where instead of "dx" it is "ds" and a different way to calculate it, but I'm not sure why? Can someone please help me to visualize this? I feel like dx would give you the infinitesimal point at any given spot between 0 to 3, so if you just calculate the circumference for everything from 0 to 3, that would give you the correct answer. Clearly, there' s a part of this that I'm not visualizing correctly, but I can't figure out what?

## \int_{0}^{3} 2 \pi x^2 dx ## // I feel since dx gives the location of the outer edge at any point, simply calculate the total of the outer edge's circumference from 0 to 3, and that would be it

Now, since I've already made this lengthy, in short, I have seen where instead of "dx" it is "ds" and a different way to calculate it, but I'm not sure why? Can someone please help me to visualize this? I feel like dx would give you the infinitesimal point at any given spot between 0 to 3, so if you just calculate the circumference for everything from 0 to 3, that would give you the correct answer. Clearly, there' s a part of this that I'm not visualizing correctly, but I can't figure out what?