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

@@ -56,8 +56,13 @@ $f(x_i)$ is the *height* of each sub-interval, and $\Delta x$ is the change in t
 ## Examples
 > Find the exact value of the integral $\int_0^1 5x \space dx$ 
 
-Relevant formula:
+Relevant formulas:
 $$ \sum_{i = 1}^n = \dfrac{(n)(n + 1)}{2} $$
+$ \Delta x = \dfrac{1 - 0$
 1. $\int_0^1 5x \space dx = \lim_{n \to \infty} \sum_{i=1}^n 5(x_i) * \Delta x$
 2. $= \lim_{n \to \infty} \sum_{i=1}^n 5(\frac{1}{n} \cdot i) \cdot \frac{1}{n}$ 
-3. $
+3. $= \lim_{n \to \infty} \sum_{i = 1}^n \dfrac{5}{n^2}\cdot i$ 
+4. $= \lim_{n \to \infty} \dfrac{5}{n^2} \sum_{i = 1}^n i$
+5. $= \lim_{x \to \infty} \dfrac{5}{n^2} \cdot \dfrac{n(n + 1)}{2}$ 
+6. $= \lim_{n \to \infty} \dfrac{5n^2 + 5n}{2n^2}$
+7. $= \dfrac{5}{2}$