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education/math/MATH1210 (calc 1)/Integrals.md
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@ -143,7 +143,7 @@ $$ \text{length of a curve} = \lim_{n \to \infty} \sum_{i=1}^{n}(\sqrt{(x_i - x_
This can also be described as:
$$ \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} $$l
$$ \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{\Delta x^2(1 + f'(x_{\hat{i}}))}$$
@ -196,3 +196,10 @@ $$ L =\int_a^b \sqrt{1 + f'(x)^2} dx$$
2. Plug into calculator
# Area Between Curves
If the area under the curve is found by approximating the space between the curve and the $x$ intercept, then the area *between* two curves can be found by approximating the space between the top curve and the bottom curve.
Visualized as a set of rectangles, each rectangle would have a corner on the top curve, and a corner on the bottom curve, with a width of $\Delta x$.
The Riemann Sum definition of the area between two curves is as follows:
$$ \lim_{n \to \infty} \sum_{i = 0}^n $$