Mathematatics

This is where one can start to expand our thinking to higher dimensions. There are two main branches of thinking. The first path uses the area under a two-dimensional curve as a slice of volume, then is rotated around the axis to create a three-dimensional object that one can find the volume of (see more information at Lumenlearning.com: 6.2 Determining Volumes by Slicing | Calculus Volume 1, 2016). The other method allows for a much more variety in the shapes solved for area and it utilizes double integrals. Just as single integrals find the area bound between one axis and a two-dimensional curve, the double integral finds the volumes bound by two axis and a three-dimensional curve.

By taking a double integral of a multivariable function, one can use that to find the volume of an object in that three-dimensional field. This sounds scary, but fret not! The first step is to understand the two variable functions. The usual notation uses x and y to solve for z: \(f(x, y)=z=r*x^{n}+t*y^{m}\) given r and t are coeffiecents and n and m are exponets for x and y. In the two dimensional cartesian plane, functions work by inputting one value (usually x) and arithmetically manipulating that to find the output (the y value). For example, input $20 to buy (get an output of) 30 candybars. Multivariable functions work very similarly. There are two inputs (x and y) which will lead to the output z. Imagine a hill. What will a person's elevation change be if they walk north 3 miles and west 1 mile. The two inputs would be distance walked north or south and the distance walked east or west. One uses the multivariable function to determine the area bound between the x and y axis. This essentially gives the height of the estimating Riemann sum rectangular prisms.

All definite integrals are bounded. To calculate single definite integrals, one is given a specific domain of x values. Double integrals also need bounds if one wants to find a definite double integrals, which will usually be given in term of x, y, or some function that combines both variables. This will be the edges of the shape and the beginning and end of the estimating Riemann sum rectangular prisms. The next step is to start applying the idea of riemann sums to volumes. Next, divide the x and y values into rectangles. Combining that with the height, one can estimate the volume of a shape under a three-dimensional curve. Using the same principle of limits, the rectangular prisms grow in number as they shrink in size. The more prisms, the more accurate the volume under the curve will be until there are infinitely many prisms for our precise value of the volume.

If you want a quick visual of 3D Riemann Sums, click here

Application and Beyond

The applications are endless. By using one of these methods, one can prove the volume formula has many three-dimensional shapes (6.2 Determining Volumes by Slicing | Calculus Volume 1, 2016). It is absolutely fascinating that from some simple rectangular areas, calculus was created and can be used to describe circular, spherical, and other oddly shaped objects.

Other uses are for finding the volume of very odd three-dimensional curves. This can be used in geology and other fields of study. When a surface is mapped, scientists can use functions to nearly perfectly replicate those surfaces. Using those created functions, they can find the volume using double integrals. This can be used to find the volume of large rock formations on earth to ginormous gaseous nebulae in the far reaches of space. These calculations are also used a great deal in physics. According to Calcworkshop.com, double integrals are used for finding center of mass, moments of inertia, charge density and probability density (Applications of Double Integrals, 2022). The principle of integration is so important that entire branches of physics are based on this concept.

The double integral is not the end. If a double integral in three dimensions is used for volume, many believe that using a triple integral in four dimensions can calculate hypervolume. Many theoretical physicists work in many higher dimensions and it is common for them to use n-th degree integrals or infinitely iterated integrals. Understanding the world requires understanding of math. While integration can be a complex monster or arithmetic, it is based on some of the simplest principles of geometry created hundreds of years ago. Mathematicians are still learning and discovering more day by day about the world and universe. All of these discoveries stem from basic concepts that are still taught in schools.