surface area

(noun)

the total area on the surface of a three-dimensional figure

Related Terms

Examples of surface area in the following topics:

  • Arc Length and Surface Area

    • Infinitesimal calculus provides us general formulas for the arc length of a curve and the surface area of a solid.
    • We will also use integration to calculate the surface area of a three-dimensional object.
    • For rotations around the $x$- and $y$-axes, surface areas $A_x$ and $A_y$ are given, respectively, as the following:
    • Now, calculate the surface area of the solid obtained by rotating $f(x)$ around the $x$-axis:
    • Use integration to find the surface area of a solid rotated around an axis and the surface area of a solid rotated around an axis

  • Cylinders and Quadric Surfaces

    • A quadric surface is any $D$-dimensional hypersurface in $(D+1)$-dimensional space defined as the locus of zeros of a quadratic polynomial.
    • The surface is formed by the points at a fixed distance from a given line segment, the axis of the cylinder.
    • The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder.
    • The surface area and the volume of a cylinder have been known since antiquity.
    • If the cylinder has a radius $r$ and length (height) $h$, then its volume is given by $V = \pi r^2h$, and its surface area is $A = 2\pi rh$ without the top and bottom, and $2\pi r(r + h)$ with them.

  • Physics and Engineering: Fluid Pressure and Force

    • 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.
    • Pressure ($p$) is force per unit area applied in a direction perpendicular to the surface of an object.
    • Mathematically, $p = \frac{F}{A}$, where $p$ is the pressure, $\mathbf{F}$ is the normal force, and $A$ is the area of the surface on contact.
    • It relates the vector surface element (a vector normal to the surface) with the normal force acting on it.
    • The total force normal to the contact surface would be:

  • Volumes of Revolution

    • The area of a ring is:
    • Summing up all of the areas along the interval gives the total volume.
    • If $g(x)=0$ (e.g. revolving an area between curve and $x$-axis), this reduces to:
    • The lateral surface area of a cylinder is $2 \pi r h$, where $r$ is the radius (in this case $x$), and $h$ is the height (in this case $[f(x)-g(x)]$).
    • Summing up all of the surface areas along the interval gives the total volume.

  • Area of a Surface of Revolution

    • A surface of revolution is a surface in Euclidean space created by rotating a curve around a straight line in its plane, known as the axis .
    • Examples of surfaces generated by a straight line are cylindrical and conical surfaces when the line is co-planar with the axis, as well as hyperboloids of one sheet when the line is skew to the axis.
    • Likewise, when the axis of rotation is the $x$-axis, and provided that $y(t)$ is never negative, the area is given by:
    • Its area is therefore:
    • Use integration to find the area of a surface of revolution

  • Line Integrals

    • More specifically, the line integral over a scalar field can be interpreted as the area under the field carved out by a particular curve.
    • This can be visualized as the surface created by $z = f(x,y)$ and a curve $C$ in the $xy$-plane.
    • The line integral of $f$ would be the area of the "curtain" created when the points of the surface that are directly over $C$ are carved out.
    • The line integral over a scalar field $f$ can be thought of as the area under the curve $C$ along a surface $z = f(x,y)$, described by the field.

  • Area Between Curves

    • Area is a quantity that expresses the extent of a two-dimensional surface or shape, or planar lamina, in the plane.
    • Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat.
    • Find the area between the two curves $f(x)=x$ and $f(x)= 0.5 \cdot x^2$ over the interval from $x=0$ to $x=2$.
    • Since $x > 0.5 \cdot x^2$ over the interval from $x=0$ to $x=2$, the area can be calculated as follows:
    • Evaluate the area between two functions using a difference of definite integrals

  • Parametric Surfaces and Surface Integrals

    • A parametric surface is a surface in the Euclidean space $R^3$ which is defined by a parametric equation.
    • A parametric surface is a surface in the Euclidean space $R^3$ which is defined by a parametric equation with two parameters: $\vec r: \Bbb{R}^2 \rightarrow \Bbb{R}^3$.
    • Parametric representation is the most general way to specify a surface.
    • The same surface admits many different parametrizations.
    • A surface integral is a definite integral taken over a surface .

  • Surfaces in Space

    • A surface is a two-dimensional, topological manifold.
    • For example, the surface of the Earth is (ideally) a two-dimensional surface, and latitude and longitude provide two-dimensional coordinates on it (except at the poles and along the 180th meridian).
    • The concept of surface finds application in physics, engineering, computer graphics, and many other disciplines, primarily in representing the surfaces of physical objects.
    • Historically, surfaces were initially defined as subspaces of Euclidean spaces.
    • In spherical coordinates, the surface can be expressed simply by $r=R$.

  • Double Integrals Over Rectangles

    • Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the $x$-axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three dimensional Cartesian plane where $z = f(x, y))$ and the plane which contains its domain.
    • The same volume can be obtained via the triple integral—the integral of a function in three variables—of the constant function $f(x, y, z) = 1$ over the above-mentioned region between the surface and the plane.
    • Double integral as volume under a surface $z = x^2 − y^2$.
    • The rectangular region at the bottom of the body is the domain of integration, while the surface is the graph of the two-variable function to be integrated.