Further Applications of Integration

Infinitesimal calculus provides us general formulas for the arc length of a curve and the surface area of a solid.

If the curve is described by the function $y = f(x) (a≤x≤b)$, the area $A_y$ is given by the integral $A_x = 2\pi\int_a^bf(x)\sqrt{1+\left(f'(x)\right)^2} \, dx$ for revolution around the $x$-axis.

Pressure is given as $p = \frac{F}{A}$ or $p = \frac{dF_n}{dA}$, where $p$ is the pressure, $\mathbf{F}$ is the normal force, and $A$ is the area of the surface on contact.

For a continuous mass distribution, the position of center of mass is given as $\mathbf R = \frac 1M \int_V\rho(\mathbf{r}) \mathbf{r} dV$ .

Calculus has broad applications in diverse fields of science; examples of integration can be found in economics and biology.

Probability density function describes the relative likelihood, or probability, that a given variable will take on a value.

A Taylor series is a representation of a function as an infinite sum of terms calculated from the values of the function's derivatives.