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education/math/MATH1210 (calc 1)/Integrals.md
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@ -84,5 +84,7 @@ To find the average value of $f(x)$ on the interval $[a, b]$ is given by the for
Average = $\dfrac{1}{b-a} \int_a^b f(x)dx$ Average = $\dfrac{1}{b-a} \int_a^b f(x)dx$
# The Fundamental Theorem of Calculus # The Fundamental Theorem of Calculus
Let $f$ be a continuous function on the closed interval $[a, b]$ and let $F$ be any antiderivative of $f$, then: 1. Let $f$ be a continuous function on the closed interval $[a, b]$ and let $F$ be any antiderivative of $f$, then:
$$\int_a^b f(x) dx = F(b) - F(a)$$ $$\int_a^b f(x) dx = F(b) - F(a)$$
2. Let $f$ be a continuous function on $[a, b]$ and let $x$ be a point in $[a, b]$.
$$ F(x) = \int_a^x f(t)dt \Rightarrow F'(x) = f(x) $$