Calculus Interpretation of Geometry
Geometry is often described as the study of shapes, and to an extent this may be true, but geometry is more than that! Geometry utilizes properties of angles, relationships with points and lines, and wholeheartedly, it is a branch of mathematics which inspires and makes students go in awe. You start by learning geometry from points, to relationships in geometric shapes, to areas and volumes, and you begin to learn real-world applications of these ideas. But, all of this is done imagining the real life scenarios given are perfect, or, in other words, the problems are not really taking into the consideration of all of the hurdles in our everyday lives. What if we are not just talking about finding areas of a regular solid, but of an object in the shape of a glass or a jar? What if we want to find lines that are not only tangent to circles but also to graphs? This may seem impossible, but it is not! Geometry’s older brother, Calculus, is a more generalized version of geometry and will help us see how we can find interesting results with this branch of Mathematics.
In this blog post, we’ll discuss about the problems we can solve in geometry and then generalize them in words Calculus can solve.
Note - this is not a place to learn Calculus (if you'd like to learn Calculus, check out Schoolhouse sessions on the topic!), but to inspire you to see that there is a lot of beauty in math, pushing you to learn about this mind-blowing subject.
Before we start, we should know about one thing- Calculus can be of two types - differential and integral calculus. Differential Calculus uses the derivative operation, whereas Integration Calculus uses the integral operation. We will look at both separately.
Lets begin with exploring Differential Calculus!
Differential Calculus Interpretation of Geometry
We’ll firstly introduce ourselves to the derivative operation notation. For a function f(x), The derivative can be written as f’(x). Note that there are many other ways to write the derivative of a function, but for the sake of this article, we will use this notation. Now let’s begin with our first idea:
Finding the value of a function
Suppose you are given a function f(x) = x^3 + 3 and you wish to calculate f(7). We can just plug in 7 to the function to obtain f(7) = 7^3 + 3 = 346. Pretty simply, right? Now, let’s see how we see this problem when solved with Calculus.
Suppose you are given a function f(x) that is not determined at x = 2 [given by the domain of f(x)] and you wish to calculate the value as the function f(x) approaches 2. This motivates us to create a new mathematical term, known as a limit. We can use properties of limits to answer the problem as## Finding the Slope
Suppose you have two points (0,1) and (2,2) and you wish to find the slope of the line. As you may have learned, the slope of the line is the difference in the y values divided by the difference in the x values. Therefore, the slope of the line that goes through the two points is (2-1)/(2-0) = 1/2.
Suppose you have a curve and you wish to find the slope of the curve. Finding the slope of the curve may not be intuitive, but a picture can best demonstrate this idea:
Notice that the red line is a tangent to the graph, and we can define the derivative of the function f(x) to be the slope of the tangent line at point (c, f(c)).
Height of a curve
Suppose we have a function f(x) and we wish to find the height of the function at a point c. What do you think the height is? Reference the above picture again to see if you can determine the height. It can be seen that the height of the graph at point c is simply what the function outputs at point c, or f(c). We have talked already about evaluation of functions, and how this does not require Calculus.
Suppose we have a function f(x) and we want to find its maximum height (or highest point). This may be simple if we have a quadratic function, but Differential Calculus can find the maximum (and minimum) value of a function using multiple techniques. We say that a function is elementary, if it basic and has no calculus-touch to it. Elementary functions include polynomials, rational functions, exponential functions, logarithmic functions, and trigonometric functions.
In general, Differential Calculus helps us understand graphs. With the help of different methods, we can sketch the graph of any elementary function (primarily via the first and second derivative test). Differential Calculus is interesting, especially when you are able to accurately graph weird looking functions, without graphing calculators!
Integral Calculus Interpretation of Geometry
Try finding the different notations of the integral. A hint is that there are two different types of integrals- definite and indefinite (or antiderivative) integrals. Now lets begin with our first idea:
Finding the Area
Suppose you want to find the area of a two dimensional figure. This is pretty straight forward as there is a formula for all shapes. For example, if we want to find the area of a rectangle with length 5 and width 2, the area is simply 5 · 2 = 10.
Suppose we have a graph f(x) and we want to find the area under the curve on the interval [a,b]. First, let’s visualize this:
Notice how this area is not like our normal geometry figure, with this region consisting of straight lines and curves. It is actually fascinating to finding the area of this region and I highly encourage you to try it out.
Finding the Length
Suppose we want to find the length of a segment with endpoints (0,0) and (3,4). This can be done simply using the distance formula to obtain 5 units. But not everything in the real world is a straight line, and most of the time the length we need to find might be of an arc or a curve. This idea is described below.
Suppose we want to find the length of an arc of the function f(x) on the interval [a,b]. We can apply integration, which can help us find useful information to construct different things. Below is a nice image which captures the arc length of a function (this is different from the arc length of a circle, don’t get confused):## Finding the Volume and Surface Area
Suppose we want to find the volume and surface of a three dimensional figure. This can be done very easily using the vast formula (and a funny thing is that the formulas are rigorously proven by calculus). For example, if I wanted to calculate the volume and surface of a cylinder with height 3 and base radius 4, we are given that the volume is πr^2h = π(4^2)(3) = 48π. Similarly, the surface area of the cylinder is 2πr^2 + 2πrh = 2π(4^2) + 2π(3•4) = 32π + 24π = 56π.
Suppose we rotate a region of a graph around a line. The solid formed has a volume and surface area that can be found by integrals. Similarly, if a three dimensional object is not uniformly dense, the volume cannot be calculated using the regular formula, as the change in density needs to be accounted for. Below is a picture of a region being rotated along a horizontal line (in this case, the x-axis).## Ending Remarks
In general, integral calculus, helps in finding the area, length, volume, and surface area of weird looking figures. You should be awestruck by the fact that finding a good estimate of your water glass or a bowling pin can be easily done by the help of Integral Calculus.
Calculus is a remarkable subject that is applied in the real world every single day! Calculus really helps us in understanding the world and in general can make the world a better place. I encourage you to take a look at all of the topics we discussed today and search on the internet for further in depth explanation. Curious minds will do remarkable things!
Thank you Hafsah M for editing this article!
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