vault backup: 2025-03-25 09:11:37

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

@@ -35,7 +35,8 @@ $$ \int f(x) dx $$
 | $\dfrac{d}{dx} \tan^{-1} x = \dfrac{1}{1+x^2}$        | $\int \dfrac{1}{\sqrt{1+x^2}}dx = \tan^{-1}x + C$        |
 | $\dfrac{d}{dx} k f(x) = k f'(x)$                      | $\int k*f(x)dx = k\int f(x)dx$                           |
 | $\dfrac{d}{dx} f(x) \pm g(x) = f'(x) \pm g'(x)$       | $\int (f(x) \pm g(x))dx = \int f(x) dx \pm \int g(x) dx$ |
-
+# Area Under a Curve
+The area under the curve $y = f(x)$ can be approximated by the equation $\sum_{i = 1}^n f(\hat{x_i})\Delta x$ where $\hat{x_i}$ is any point on the interval $[x_{i - 1}, x_i]$, and the curve is divided into $n$ equal parts of width $\Delta x$
 # Definite Integrals
 Let $f$ be a continuous function on the interval $[a, b]$. Divide $[a, b]$ into $n$ equal parts of width $\Delta x = \dfrac{b - a}{n}$ . Let $x_0, x_1, x_2, \cdots, x_3$ be the endpoints of the subdivision.
 
@@ -43,4 +44,6 @@ The definite integral of $f(x)$ with respect to $x$ from $x = a$ to $x = b$ can
 $$ \int_{a}^b f(x) dx $$
 
 And __can__ be defined as:
-$$ \int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i = 1}^n f(x_i)\Delta x$$
+$$ \int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i = 1}^n f(x_i)\Delta x$$
+
+$f(x_i)$ is the *height* of each sub-interval, and $\Delta x$ is the change in the *x* interval, so $f(x_i) \Delta x$ is solving for the area of each sub-interval.