Volume is the measure of the amount of space occupied by a three-dimensional object.
The units of volume depend on the units used to measure the dimensions of the object. For example, if the dimensions are measured in meters, the volume will be expressed in cubic meters (m³).
Integration is commonly used in engineering, physics, and architecture to calculate the volume of irregularly shaped objects and to determine the amount of material needed for construction projects. It is also used in fluid mechanics to calculate the volume of fluids in containers or pipes.
A definite integral is used to find the exact volume of a three-dimensional object, while an indefinite integral is used to find the general formula for the volume. In other words, a definite integral gives a specific numerical value for the volume, while an indefinite integral gives a function that can be used to find the volume for any given set of dimensions.
The method of slicing involves dividing the three-dimensional object into thin slices and summing the volumes of each slice using integration. The shell method, on the other hand, involves summing the volumes of cylindrical shells that make up the object. The choice of method depends on the shape of the object and which method is easier to set up for integration.