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education/math/MATH1210 (calc 1)/Integrals.md
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@ -72,8 +72,13 @@ $$ \Delta x = \dfrac{1 - 0}{n} = \dfrac{1}{n}$$$$ x_i = 0 + \Delta xi + \dfrac{1
7. $= \dfrac{5}{2}$ 7. $= \dfrac{5}{2}$
# Properties of Integrals # Properties of Integrals
1. $\int_a^a f(x)dx = 0$ - An integral with a range of zero will always evaluate to zero. 1. $\int_a^a f(x)dx = 0$ - An integral with a domain of zero will always evaluate to zero.
2. $\int_b^a f(x)dx = -\int_a^b f(x) dx$ - The integral from $a \to b$ is equal to the integral from $-(b\to a)$ 2. $\int_b^a f(x)dx = -\int_a^b f(x) dx$ - The integral from $a \to b$ is equal to the integral from $-(b\to a)$
3. $\int_a^b cf(x) dx = c \int_a^b f(x) dx$ - A constant from inside of an integral can be moved outside of an integral 3. $\int_a^b cf(x) dx = c \int_a^b f(x) dx$ - A constant from inside of an integral can be moved outside of an integral
4. $\int_a^b f(x) \pm g(x) dx = \int_a^b f(x) dx \pm \int_a^b g(x)dx$ - Integrals can be distributed 4. $\int_a^b f(x) \pm g(x) dx = \int_a^b f(x) dx \pm \int_a^b g(x)dx$ - Integrals can be distributed
5. $\int_a^c f(x)dx = \int_a^b f(x)dx + \int_b^c f(x)dx$ - An integral can be split into two smaller integrals covering the same range, added together. 5. $\int_a^c f(x)dx = \int_a^b f(x)dx + \int_b^c f(x)dx$ - An integral can be split into two smaller integrals covering the same domain, added together.
# Averages
To find the average value of $f(x)$ on the interval $[a, b]$ is given by the formula:
Average = $\dfrac{1}{b-a} \int_a^b f(x)dx$