# Integration, Differentiation and Determinants

Differentiation and Integration are the two most fundamental concepts in Calculus. Both differentiation and integration are used to measure change. Differentiation is in a way, the direct opposite or the inverse of integration.

Differentiation is the rate of change of an algebraic function, while Integration is the accumulation of an algebraic function over a range. Integration is divided into definite and indefinite integration.

In indefinite integration, the algebraic function can be combined infinite times. In definite integration, the algebraic function can only be combined for a limited number of times. In other words, there are upper and lower limits in a definite integration. In an indefinite integration, there are no upper or lower limits.

The real life applications of differentiation and integration can be found across a multitude of streams including Engineering, Statistics, Medical Science, Business and Architecture.

To illustrate, let’s look at some examples. An apple that falls from the tree increases in speed as it reaches the floor.

But how do you calculate the continuous rate of change in velocity overtime? This is where differentiation comes in.

**Important properties of determinants**

- Reflection property: The determinant remains unchanged if its rows are converted into columns and columns into rows. This is known as the property of reflection.
2. Property of all zeros: If all elements of a row (or column) are zero, the determinant is zero.

3. Proportionality (Repetition) Properties: If all elements of a row (or column) are proportional (same) to the elements of another row (or column), the determinant is zero.

4. Change property: The exchange of two rows (or columns) of the determinant changes its sign.

5. Multiple scalar property: If all elements of a row (or column) of the determinant are multiplied by a non-axial constant, the determinant is multiplied by the same constant.

Using differential calculus, one can break the problem into smaller steps and solve it through simple equations. In integration, we reverse the process to find the sum of an infinitesimal number of varying elements.

To put that into perspective, consider a swimming pool with artificial waves. If you are tasked with finding the surface height of the water in the pool, you will have to consolidate all the variations in the water surface caused by the waves.

To accomplish this complicated task, one could accumulate the various calculations together and thereby sum up the surface area through integral calculus.

Bringing it back to mathematical terms, if a differentiation is the slope or gradient of a graph at a given point, then integration is the area under the curve.

A determinant is the added product of all the elements in a square matrix. Determinants arise when mathematicians are trying to solve different linear equations at the same time.

Also, the determinant of a matrix is used to find its inverse. Differentiation integration of determinant is an advanced concept that brings all the previously discussed mathematical ideas together.

For scientists and industries tasked with complex and dynamic problems, differentiation and integration of determinants is very helpful.

In some cases, it provides a way to measure the maximum cumulative accuracy and efficiency that is achievable under the given variables.