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education/math/MATH1210 (calc 1)/Integrals.md
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@ -135,4 +135,12 @@ With a series of points on a curve we can find the distance between each point.
As we increase $n$, the precision of which the curve is estimated increases. As we increase $n$, the precision of which the curve is estimated increases.
This means that: This means that:
$$ len(curve) = \lim_{n \to \infty} \sum_{i=1}^{n}(\p{}$$ $$ \text{length of a curve} = \lim_{n \to \infty} \sum_{i=1}^{n}(\text{length of line segment)}$$
Using the distance formula, we know that the length of the line segment can be found with:
$$ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$
1. So the entire equation is:
$$ \text{length of a curve} = \lim_{n \to \infty} \sum_{i=1}^{n}(\sqrt{(x_i - x_{i-1})^2 + (y_i - y_{i-1})^2}) $$
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:
$$ \lim_{n \to \infty} \sum_{i = 1}^n) $$