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education/math/MATH1210 (calc 1)/Integrals.md
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@ -143,4 +143,10 @@ $$ \text{length of a curve} = \lim_{n \to \infty} \sum_{i=1}^{n}(\sqrt{(x_i - x_
This can also be described as:
$$ \text{length of a curve} = \lim_{n \to \infty} \sum_{i=1}^{n}(\sqrt{(\Delta x)^2 +(\Delta y)^2}) $$
2. Using the mean value theorem:
$$ \lim_{n \to \infty} \sum_{i = 1}^n) $$
$$ \lim_{n \to \infty} \sum_{i = 1}^n)\sqrt{\Delta x^2 + (F(x_i) - F(x_{i-1}))i^2} $$
$$ \lim_{n \to \infty} \sum_{i=1}^n \sqrt{\Delta x ^2 + (f'(x_{\hat{i}}))(x_i - x_{i-1})^2}$$
3. Factoring out $\Delta x$
$$ \lim_{n \to \infty} \sum_{i=1}^n \sqrt{x^2(1 + f'(\Delta x_{\hat{i}}))}$$
4. Moving $\Delta x$ out of the root
$$ \lim_{n \to \infty} \sum_{i=1}^n \sqrt{x^2(1 + f'(\Delta x_{\hat{i}}))}$$