Applications: volume

Understanding Volume

Volume is the amount of space an object occupies and can be found by integrating the cross-sectional area along an axis.
Volume is measured in cubic units like cubic metres (m³), cubic centimetres (cm³), etc.
For a three-dimensional object, the volume is obtained by integrating the area function over the specified dimensions of the object.

Volume of Common Shapes

The volume of a rectangular prism is found by multiplying the length, width, and height: V = lwh.
For a cylinder, volume is found by multiplying the area of the base (a circle) by the height: V = πr²h.
The volume of a cone is found by multiplying the area of the base (a circle) by one third of the height: V = (1/3)πr²h.
A sphere’s volume is calculated by 4/3 πr³.

Volume of Revolution

If a shape is rotated about one of the axes, it creates a solid of revolution.
The volume of this solid is found by disc method or shell method.
The disc method involves adding up the volumes of all the tiny discs that make up the solid (integral of πr²dx), where r is the distance from the axis of rotation.
The shell method involves adding up the volumes of all the tiny cylinders (“shells”) that make up the solid (integral of 2πrhdx), where r is the radius of the cylinder and h is its height.

Integration and Volume

Integration in calculus can be used to calculate the volume of an irregular shape.
This is usually done by slicing the solid into thin pieces, estimating the volume of each slice, and then summing these volumes.
As the thickness of the slices approaches zero, the estimate becomes an exact calculation, which is the definite integral.

Understanding these principles will enable you to calculate the volume of regular and irregular solids. Practice is crucial in mastering volume calculation, as it involves understanding shapes and their properties, as well as applying integration techniques.