Mathematical methods have a wide scope in almost every field, be it Economics, Physics, Geography, or any other. Detailed knowledge and correct usage of Surface Area and Volume are important to excel and achieve perfection.

Both concepts become significant while solving real-life problems relating to measurements and are studied under the Mensuration unit. Methods of Integration find applications in the calculation of Area and Volume of irregular and complex surfaces.

**Surface Area vs Volume**

The difference between Surface Area and Volume is that the Surface Area measures the area occupied by the uppermost layer of a surface or put differently it is the area of all the shapes/planes that make up the figures/solids while Volume is the measure of carrying capacity of a figure/shape or the space enclosed within the figure.

## Comparison Table Between Surface Area and Volume (in Tabular Form)

Parameter of Comparison | Surface Area | Volume |
---|---|---|

Definition | It is the area of all the shapes/planes that constitute the uppermost layer of a figure/solid. | It is the space contained in the 3-D solid/figure or the amount of air inside it. |

Dimension | It is a 2-Dimensional concept. The answer is always in a unit square like m² or cm². | It is a 3-Dimensional concept. The answer is always in a unit cube like m³ or cm³. |

Calculated for? | Surface Area can be calculated for any figure in the plane or in space. | Volume are calculated for solids only because they have 3 dimensions. |

Real-life examples | We find the surface area to estimate the area of walls to be painted to calculate the costs. | We find Volume to estimate how many goods can be kept in a store. |

Methods to calculate | By integration using the arc or the revolution of arc concept for complex figures/solids. | By integration using the disk method, washer method, or cylindrical shells methods. Some formulas are special cases of the method as in: For cube = S*S*S |

Some formulas are predetermined as in: For Square= S*S and Sphere=4πr² |

## What is Surface Area?

Surface Area is the total area covered by the surface. If we convert our surface into a 2-D Plane and then calculate the total area, we get the Surface Area. It can be calculated for any figure, for a line segment that is one-dimensional, the surface area is zero.

We’ll always have positive values as the area is a scalar and has magnitude only. Whatever be the dimension of surface, the area has two-dimensions and Hence, it would have units like m² or cm² or mm².

It is widely used concepts by Architects and is very important and useful for even common man. For example, to estimate time, speed or cost of painting walls, or for laying down fences or to delimit the constituencies, etc.

Some Formulas:

- Square : S*S
- Rectangle : L*B
- Sphere. : 4πr²
- Cone. : πr(l+r)

Several Methods to find Area of complex figures were formulated: The method to find the Surface Area is to visualize the solid or 3-D object as a revolution of a plane curve. For example, we can generate a sphere by revolving a semi-circle. In this case, the area is sum total of all curved surface Area of very small cylindrical pieces that can be cut. Here, is when integration comes to play; area equals integration of 2πf(x)√(1+(f'(x))²) with respect to x from x=a to x=b.

## What is Volume?

Volume is the carrying capacity or the amount of air contained inside a solid/figure. It can be calculated for figures that have more than 2 dimensions.

We’ll have positive values of volume because it is a scalar and has magnitude only. The Volume is 3-Dimensional and Hence, it would have units like m³ or mm³ or cm³.

It is widely used in businesses to estimate storage capacity and in scientific equipment like beakers, syringes, etc. For example, to store grain sacks or to measure medicine.

Some Formulas:

- Cube : S*S*S
- Cuboid : L*B*H
- Sphere. : ( 4/3) πr³
- Cone. : (1/3)πr²h

Methods to Calculate Volume of complex and irregular figures:

**Volume by slicing:**If the cross-sectional area of a solid is known, we can find the volume by integrating the area as a function of a variable for the domain of the variable.**Volume by disks:**By visualizing the solids as a revolution of a plane figure. We can then estimate the cross-sectional area of the small and small pieces of the solid. The volume would be the integration of π(f(x))² with respect to x for the domain of x.**Volume by washers:**In this case, our solid of revolution is formed by a region between two planes/curves. The cross-sectional area would be washer shaped and Volume would be the integration of π[(f(x))²- (g(x))²] with respect to x for the domain of x.**Volume by Cylindrical Shells:**We can also solve the above problems without calculating the area of the cross-section by visualization of our solid as a body of encircled very thin cylinders. The Volume is the integration of 2πxf(x) with respect to x for the range of x.

**Main Differences Between Surface Area and Volume**

- The Surface Area is a sum total of Area of the planes that form a surface/shape while Volume is the space enclosed within a figure/shape/surface.
- The Surface Area is a 2-Dimensional concept with units m², cm² or mm² whereas Volume is a 3-Dimensional concept with m³, cm³ or mm³ as units.
- Surface Area can be found for 2-D figures like Circle, Square, Rectangle but Volume cannot be found for them. While both can be found for 3-D solids/figures like Cube, Sphere, Cylinder, or Cone.
- Surface Area is found for estimating area of walls to be painted while Volume is found to estimate storage capacity within walls.
- The area is calculated by integrating the arc or the revolution of an arc (depending on the figure) while Volume is calculated by integrating the revolution of a surface. These methods are used while considering very complex functions and are a part of higher-level studies.

## Conclusion

It is very important for everyone to distinguish between the concepts. The Surface Area is the total area of the uppermost layer of a surface or the area of all the planes that constitute the figure by their intersection and Volume is the amount of air that can be filled or enclosed within the space between the intersection of such planes.

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