https://youtu.be/4bMEIf6WD8M

In this video I go over further into the wonderful world of calculus with parametric curves and this time go over part 1 of the proof for the formula for the surface area of a shape formed by rotating a curve about the x-axis. In part 1 I look at the case where the parametric equations can also be written as a typical function, y = F(x), which is the same as the surface area proof that I covered in my earlier video. In this case, we can simply use the substitution rule for definite integrals to change the basic surface area integral formula to one that accounts for the extra parametric parameter, just as in my earlier videos on arc length for parametric curves. Also just like for arc length, this formula derived using substitution is also valid even if the parametric equations can’t be written in the form of y = F(x) (i.e. a one-to-one function). I will illustrate this in part 2 by using polygonal approximation, again just like as for arc length, so stay tuned for that video!

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**Surface Area**

Recall from my earlier video that the surface area of a shape formed by rotating a function about the x-axis is given by the formula:

If F is positive and has a continuous derivative, we define the surface area of the surface obtained by rotating the curve y = F(x), a ≤ x ≤ b, about the x-axis as:

![](https://steemitimages.com/DQmNYMkwh4KPG4eBZYtaw4fZ6WVXmPEoMmMWLTWfsnbJF8X/image.png)

If instead y = F(x) was written in parametric form, then we can use the Substitution Rule for Definite Integrals to obtain:

![](https://steemitimages.com/DQmc1GfvzH7jcHPj4nWFrymeLjTdVtJ1AQtn4Y75b2szEZP/image.png)

![](https://steemitimages.com/DQmaAsQy9cL1NF4aLYHF14rHHYbbjT5Wwdztij6dSzX2oVr/image.png)

Note: This derivation is for the case where the parametric equations x = f(t) and y = g(t) can be written in the form y = F(x) (i.e. a one-to-one function).

But in fact this formula is still valid even if the parametric curve can't be written as y = F(x). 

![](https://steemitimages.com/DQmfBZrK5mf8bWDoVStUCcvgwBNigYgXrHMY1vUsyPtoGV8/image.png)

I will go over this in Part 2 so stay tuned!