Area
Find the area of the region enclosed by a curve $y=f(x)$ from $a$ to $b$ where $f$ is continuous and positive for all $x\in[a,b]$.
\[\text{A} = \int_a^b dA = \int_a^bf(x)dx\]where $dA=f(x)dx$ represents a tiny rectangle under the curve with the size of length $f(x)$ and width $dx$.
Arc length
Find the arc length of a curve $y=f(x)$ from $a$ to $b$.
\[\text{Arc Legnth} = \int_a^b dL = \int_a^b \sqrt{1+f'(x)^2}dx\]where $dL=\sqrt{dx^2+dy^2} = \sqrt{1+\left(\frac{dy}{dx}\right)^2}$ is the length of a tiny line segment from the point $(x,f(x))$ to $(x+dx,f(x+dx))$.
Volume of a solid
The disk method
Find the volume of the solid of revolution generated when the finite region $S$ that lines between $x=a$, $x=b$, and $y=f(x)$ is revolved about $x$-axis.
\[V=\int_{a}^bdV = \int_a^b f(x)^2\pi dx.\]The washer method
Find the volume of the solid of revolution generated when the finite region $S$ that lines between $x=a$, $x=b$, $y=f(x)$, and $y=g(x)$ with $f(x)>g(x)$ on $[a,b]$ is revolved about $x$-axis.
\[V=\int_{a}^bdV = \int_a^b \left(f(x)^2-g(x)^2\right)\pi dx.\]The cylinder method
Find the volume of the solid of revolution generated when the finite region $S$ that lines between $x=a$, $x=b$, and $y=f(x)$ is revolved about $x=c$ with $b>a>c$.
\[V=\int_a^bdV = \int_a^b2 (x-c)\pi f(x)dx.\]Center of Mass
Consider two particles with masses $m_1$ and $m_2$, located at positions $x_1$ and $x_2$, respectively. The center of mass for these two particles is given by:
\[\frac{m_1x_1 + m_2x_2}{m_1 + m_2}\]For a system of $n$ particles, the center of mass can be determined using induction:
\[\text{Center of mass} = \frac{\sum_{i=1}^n m_i x_i}{\sum_{i=1}^n m_i}\]Applying this formula to a region with uniform density $\delta$, bounded by the curves $y = f(x)$, $x = a$, $x = b$, and $y = 0$, where $f(x)$ is positive and integrable, yields the coordinates of the center of mass $(\bar{x}, \bar{y})$:
\[\bar{x} = \frac{\int_{a}^b x\delta f(x) \, dx}{\int_a^b f(x) dx}\] \[\bar{y} = \frac{\int_a^b \frac{\delta}{2}f^2(x) \, dx}{\int_a^b f(x) dx}\]