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education/math/MATH1210 (calc 1)/Integrals.md
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@ -143,10 +143,18 @@ $$ \text{length of a curve} = \lim_{n \to \infty} \sum_{i=1}^{n}(\sqrt{(x_i - x_
This can also be described as:
$$ \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)\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_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}}))}$$
$$ \lim_{n \to \infty} \sum_{i=1}^n \sqrt{\Delta x^2(1 + f'(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}}))}$$
$$ \lim_{n \to \infty} \sum_{i=1}^n \sqrt{(1 + f'(x_{\hat{i}}))} \Delta x$$
5. As an integral:
$$ L =\int_a^b \sqrt{1 + f'(x)^2} dx$$
## Examples
> Find the length of the curve $y = -\frac{5}{12}x + \frac{7}{12}$ from the point $(-1, 1)$
1. $L = \int_{-1}^8 \sqrt{1 + (-\frac{5}{12})^2} dx$
2. $= \int_{-1}^8 \sqrt{1 + \frac{25}{144}} dx$
3. = $\int_{-1}^8 \sqrt{\frac{169}{144}}