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

## Without Calculus

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 simple, right? Now, let’s see how we see this problem when solved with Calculus.

## Finding the Slope

## Without Calculus

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.

## Height of a Curve

## Without Calculus

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.

## Ending Remarks

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

## Without Calculus

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.

## With Calculus

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

## Without Calculus

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.

## Finding the Volume and Surface Area

## Without Calculus

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$^2$

h = 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.

## 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!