The difference between the top and bottom curves gives the height of the infinitesimal rectangles.

To find the area of a region bounded by a polar curve $r = f(\theta)$ between angles $\theta = a$ and $\theta = b$, we integrate the area of infinitesimal circular sectors ($A = \frac{1}{2}r^2 \theta$).

The length of a polar curve $r = f(\theta)$ from $\theta = a$ to $\theta = b$ is derived from the standard arc length formula by treating $\theta$ as the parameter.

Where $R(x)$ is the radius of the disk, measured from the axis of revolution to the curve boundary.

Where $R(x)$ is the outer radius (distance from axis to the farthest curve) and $r(x)$ is the inner radius (distance from axis to the closest curve, defining the hole).

Where $r(x)$ is the radius of the shell (often just $x$), and $h(x)$ is the height of the shell (the function value). Note that when rotating around the y-axis, the Shell Method integrates with respect to $x$.

Provided that $f'(x)$ is continuous on the interval $[a, b]$.

If the rotation is around the y-axis, the radius function changes from $f(x)$ to $x$.

If the region is bounded between a top curve $f(x)$ and a bottom curve $g(x)$, the moment formulas adjust to calculate the properties of the enclosed slice:

• $M_y = \int_a^b x [f(x) - g(x)] \, dx$
• $M_x = \frac{1}{2} \int_a^b \left( [f(x)]^2 - [g(x)]^2 \right) \, dx$

These calculate resistance to rotation about the primary axes.

To find the geometric centroid $(\bar{x}, \bar{y})$ of a curve $y = f(x)$ from $x=a$ to $x=b$, you treat the curve as a thin wire of uniform density. The total mass is replaced by the arc length $L$, and the moments are calculated using the differential arc length $ds = \sqrt{1 + [f'(x)]^2} \, dx$.

Common applications include compressing a spring (Hooke's Law: $F(x) = kx$) or pumping liquid out of a tank. When pumping liquid out of a tank, the "object" being moved is a horizontal slice of water, and the "distance" is how far that specific slice must be lifted to reach the top.

Where $\rho$ is the fluid density, $g$ is gravity, $h(y)$ is the depth of the strip at $y$, and $w(y)$ is the width of the strip at $y$.

The Center of Pressure ($y_{cp}$) is the point where the total hydrostatic force is assumed to be concentrated to produce the same moment as the distributed fluid pressure. It is found by dividing the moment of the fluid force by the total fluid force.

$y_{cp} = \frac{\int_a^b y \cdot dF}{\int_a^b dF} = \frac{\int_a^b y \cdot \rho g \cdot h(y) \cdot w(y) \, dy}{F}$

The surface area $A$ of a surface of revolution generated by rotating a plane curve of length $L$ about an external axis in its plane is equal to the product of the arc length $L$ and the distance traveled by its geometric centroid. Where $\bar{r}$ is the perpendicular distance from the axis of rotation to the centroid of the curve.

The volume $V$ of a solid of revolution generated by rotating a plane region of area $A$ about an external axis in its plane is equal to the product of the area $A$ and the distance traveled by its geometric centroid. Where $\bar{r}$ is the perpendicular distance from the axis of rotation to the centroid of the area.

• Consumer Surplus: The monetary gain obtained by consumers because they are able to purchase a product for a price that is less than the highest price they would be willing to pay. Given a demand curve $D(x)$ and an equilibrium point $(X, P)$.
• Producer Surplus: The amount producers benefit by selling at a market price that is higher than the least they would be willing to sell for. Given a supply curve $S(x)$.

In probability, a continuous random variable $X$ is described by a Probability Density Function (PDF), $f(x)$. The probability that $X$ falls between values $a$ and $b$ is found by integrating the PDF. By definition, the total area under any PDF from $-\infty$ to $\infty$ must equal exactly 1.