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education/math/MATH1210 (calc 1)/Integrals.md

@@ -107,6 +107,9 @@ This formula can also be stated as $\int_a^b f(x)dx = f(c)(b-a)$
 This theorem tells us that a continuous function on the closed interval will obtain its average for at least one point in the interval.
 
 # U-Substitution
+When you see $dx$ or $du$ in a function, it can be thought of as roughly analogous to $\Delta x$, where the change in $x$ is infinitesimally small.
+
+Thinking back to derivatives, when solving for $\dfrac{dy}{dx}$, you're solving for the rate of change of $y$ (across an infinitely small distance) over the rate of change of $x$ (across an infinitely small i$nstance). Given that the *slope* of a line is described as $\dfrac{\text{rise}}{\text{run}}$, 
 ## Formulas
 - $\int k {du} = ku + C$
 - $\int u^n du = \frac{1}{n+1}u^{n+1} + C$
@@ -204,4 +207,10 @@ The height of the rectangle, or the distance between the curves at a given point
 
 The Riemann Sum definition of the area between two curves is as follows:
 
-$$ \lim_{n \to \infty} \sum_{i = 0}^n (f(x_i)-g(x_i)\cdot \Delta x)$$
+$$ \lim_{n \to \infty} \sum_{i = 1}^n (f(x_i)-g(x_i)\cdot \Delta x)$$
+- $i$ is the sub-interval
+- $x_i$ is the $x$ coordinate at a given sub-interval
+- $\Delta x$ is the width of each sub-interval.
+
+This sum can also be described as:
+$$ = \int_a^b $$