Consider two continuous functions, $f(x)$ and $g(x)$, such that $f(x) \ge g(x)$ for all $x$ in the interval $[a, b]$. The area $A$ of the region bounded by the graphs of $y = f(x)$, $y = g(x)$, and the vertical lines $x = a$ and $x = b$ is given by the definite integral: $A = \int_{a}^{b} [f(x) - g(x)] dx$ If the curves intersect, the limits of integration $a$ and $b$ are typically the x-coordinates of these intersection points. If the region is bounded by curves $x = p(y)$ and $x = q(y)$ from $y = c$ to $y = d$, where $p(y) \ge q(y)$, the area is given by: $A = \int_{c}^{d} [p(y) - q(y)] dy$

1. Disk Method: If a region bounded by $y = f(x)$, the x-axis, $x = a$, and $x = b$ is revolved around the x-axis, the volume $V$ is given by: $V = \pi \int_{a}^{b} [f(x)]^2 dx$ If revolved around the y-axis, and the region is bounded by $x = g(y)$, the y-axis, $y = c$, and $y = d$, the volume is: $V = \pi \int_{c}^{d} [g(y)]^2 dy$ 2. Washer Method: If a region between $y = f(x)$ (outer radius) and $y = g(x)$ (inner radius) is revolved around the x-axis from $x = a$ to $x = b$, the volume is: $V = \pi \int_{a}^{b} [[f(x)]^2 - [g(x)]^2] dx$ 3. Shell Method: If a region bounded by $y = f(x)$, the x-axis, $x = a$, and $x = b$ is revolved around the y-axis, the volume is: $V = 2\pi \int_{a}^{b} x f(x) dx$

The arc length $L$ of a curve defined by $y = f(x)$ from $x = a$ to $x = b$ is given by the integral: $L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$ If the curve is defined parametrically by $x = x(t)$ and $y = y(t)$ for $t_1 \le t \le t_2$, the arc length is: $L = \int_{t_1}^{t_2} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$

When a force $F(x)$ varies with position $x$, the work $W$ done by this force in moving an object from position $a$ to position $b$ along a straight line is given by the definite integral: $W = \int_{a}^{b} F(x) dx$ This concept is fundamental in physics, for example, in calculating the work done to stretch or compress a spring (Hooke's Law, $F = -kx$).

The average value of a continuous function $f(x)$ over the interval $[a, b]$ is defined as: $f_{avg} = \frac{1}{b-a} \int_{a}^{b} f(x) dx$ This formula essentially divides the total 'area' under the curve by the 'width' of the interval to find an equivalent constant height.