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education/math/MATH1220 (calc II)/Integral Review.md
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@ -23,7 +23,3 @@ $$ \sum_{i = 1}^n i = \frac{n(n+1)}{2}$$
$$ \int_0^1 x^2 dx = \frac{1}{3} x^3 \Big |_0^1 = 1/3(1^3)- 1/3 (0)^3 = 1/3$$ $$ \int_0^1 x^2 dx = \frac{1}{3} x^3 \Big |_0^1 = 1/3(1^3)- 1/3 (0)^3 = 1/3$$
2. Find the Riemann Sum under the curve between -2 and 2 of the function $2x + 2$. 2. Find the Riemann Sum under the curve between -2 and 2 of the function $2x + 2$.
$$ \int_{-2}^2 (2x + 2)dx = \lim_{n \to \infty} \sum_{i = 1}^nf(x_i)\Delta x $$ $$ \int_{-2}^2 (2x + 2)dx = \lim_{n \to \infty} \sum_{i = 1}^nf(x_i)\Delta x $$
> Using the fact that $x_i = \Delta xi + a$, $f(x_i) = 2$
$$ = lim_{n \to \infty} \sum_{i=1}^n(2x_i + 2)\frac{4}{n}$$
$$ = \lim_{n \to \infty} \sum_{i = 1}^n (2(-2 +\frac{4i}{n}) + 2)\frac{4}{n}$$
$$ = \lim_{n \to \infty} \sum_{i = 1}^n(-4 + \frac{8i}{n} + 2)\frac{4}{n} $$