Examples of surface area in the following topics:
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- Cell size is limited in accordance with the ratio of cell surface area to volume.
- Therefore, as a cell increases in size, its surface area-to-volume ratio decreases.
- However, increased surface area can cause problems as well.
- Notice that as a cell increases in size, its surface area-to-volume ratio decreases.
- The cell on the left has a volume of 1 mm3 and a surface area of 6 mm2, with a surface area-to-volume ratio of 6 to 1, whereas the cell on the right has a volume of 8 mm3 and a surface area of 24 mm2, with a surface area-to-volume ratio of 3 to 1.
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- 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
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- Recall that any three-dimensional object has a surface area and volume; the ratio of these two quantities is the surface-to-volume ratio.
- Consider a cell shaped like a perfect sphere: it has a surface area of 4πr2, and a volume of (4/3)πr3.
- The larger the size of the sphere, or animal, the less surface area for diffusion it possesses.
- Surface-to-volume ratio also applies to other areas of animal development, such as the relationship between muscle mass and cross-sectional surface area in supporting skeletons or in the relationship between muscle mass and the generation of dissipation of heat.
- The surface area of a sphere is 4πr2 and it has a volume of (4/3)πr3 which makes the surface-to-volume ratio 3/r.
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- 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.
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- Pressure is scalar quantity which is defined as force per unit area where the force acts in a direction perpendicular to the surface.
- where p is pressure, F is the force acting perpendicular to the surface to which this force is applied, and A is the area of the surface.
- It has an area of contact (with the surface upon which it is resting) of 0.1 m2, thus exerting a pressure of 1,000 Pa on that surface.
- That same block in a different configuration (also in Figure 2), in which the block is placed vertically, has an area of contact with the surface upon which it is resting of 0.01 m2, thus exerting a pressure of 10,000 Pa—10 times larger than the first configuration due to a decrease in the surface area by a factor of 10.
- Alternatively, an object having a weight larger than another object of the same dimensionality and area of contact with a given surface will exert a greater pressure on that surface due to an increase in force.
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- Electric flux is the rate of flow of the electric field through a given area.
- Electric flux is the rate of flow of the electric field through a given area (see ).
- If the electric field is uniform, the electric flux passing through a surface of vector area S is $\Phi_E = \mathbf{E} \cdot \mathbf{S} = ES \cos \theta$ where E is the magnitude of the electric field (having units of V/m), S is the area of the surface, and θ is the angle between the electric field lines and the normal (perpendicular) to S.
- For a non-uniform electric field, the electric flux dΦE through a small surface area dS is given by $d\Phi_E = \mathbf{E} \cdot d\mathbf{S}$ (the electric field, E, multiplied by the component of area perpendicular to the field).
- Gauss' Law describes the electric flux over a surface S as the surface integral: $\Phi_E = \iint_S \mathbf{E} \cdot d\mathbf{S}$ where E is the electric field and dS is a differential area on the closed surface S with an outward facing surface normal defining its direction.
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- 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:
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- Surface tension is a contractive tendency of the surface of a liquid that allows it to resist an external force.
- If no force acts normal (perpendicular) to a tensioned surface, the surface must remain flat.
- But if the pressure on one side of the surface differs from pressure on the other side, the pressure difference times the surface area results in a normal force.
- Surface tension is expressed in units of force per unit length or of energy per unit area (for instance, N/m or J/m2).
- The two are equivalent, but when referring to energy per unit area, people use the term "surface energy," which is a more general term in the sense that it applies to solids as well as to liquids.
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- Attractive forces between molecules cause effects such as surface tension and capillary action.
- Surface tension is a contractive tendency of the surface of a liquid that allows it to resist an external force.
- This creates some internal pressure and forces liquid surfaces to contract to the minimal area.
- Surface tension has the unit of force per unit length, or of energy per unit area.
- However, when we refer to energy per unit of area, we use the term surface energy, which is more general in that it applies to solids as well as liquids.
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- Surface tension is the tendency of a liquid surface to resist forces applied to it.
- This effect is a result of cohesion of the molecules of the liquid causing the surface of the liquid to contract to the smallest area possible.
- Where the surfaces meet, forces must be in equilibrium.
- The leaf is a hydrophobic surface.
- Summarize the cause for different surface tensions at a liquid's surface