In this post I’ll be revealing how you can derive the formula which can be used to **find areas underneath curves**, from absolute scratch. Now, just below, what you will find is the diagram that will help us produce this formula…

**In this diagram what you will discover is that:**

*Please read the following contents carefully…

- A length
**a**exists, which starts at the origin**O**and ends at**a**; - A length
**x**exists, which starts at the origin**O**and ends at**x**; - A length
**x+𝛿**exists, which starts at the origin**x****O**and ends at;**x+𝛿x** - A length
**𝛿x**exists, which starts at**x**and ends at**x+𝛿x**; - A height
**y**exists, which starts at the origin**O**and ends at**y**; - A height
**y+𝛿y**exists, which starts at the origin**O**and ends at**y+𝛿y;** - A height
**𝛿y**exists, which starts at**y**and ends at**y+𝛿y;** - There is a curve called
**y=f(x)**; - There is an area underneath the curve called
**A**which commences at**a**and ends at**x**; - There is an area underneath the curve called
**𝛿A**which commences atand ends at**x**(Note: If you extend the distance from a to x what you get is a larger area, and the change in area can be measured. This change or difference is called**x+𝛿****x****𝛿****A**); - There is a rectangle that exists called
**QRUT**. It has an area which is**y𝛿x**; - There is a rectangle that exists called
**PRUS**. It has an area which is**(y+𝛿y)𝛿x**; **𝛿A**has an area larger than that of the rectangle**QRUT**, but smaller than that of the rectangle**PRUS**.

**Producing the formula with the information we’ve discovered…**

Ok, so we want to produce the formula which will help us find areas underneath curves from absolute scratch. At our disposal we have a helpful diagram (which we’ve looked at and analysed carefully) and we’ve been able to discover a few facts about it. I think we can now get to work…

Let’s start off by saying that:

**Area QRUT < 𝛿A < Area PRUS**

Which is something we already discovered.

If this is the case, we can say that:

**y𝛿x < 𝛿A < (y+𝛿y)𝛿x**

Now, check out what happens when we divide all the elements of this expression by **𝛿x**:

What we end up with is…

Alright, now you may be saying to yourself, why do I need to know this? Well, it turns out that:

This is because as **𝛿x **approaches** 0**, **𝛿y **approaches** 0 **leaving** (𝛿A)/(𝛿x) **sandwiched between **y** and **y+0.000000000000000001** which is virtually y.

And, also…

As a consequence, this ultimately means that:

This is incredibly significant, because if we then integrate both sides of this equation, we get:

And…

Now, this equation can actually be used to find the area **A** underneath the curve from a to x. What we’re basically saying is that this area is equal to some function of x plus a constant. This ‘some function of x’ occurs when we integrate y which is a function of x.

**Finalising the formula…**

Alright so we’ve managed to latch on to something incredibly significant… We’ve got an important equation:

However, it is not complete. We need to know what the constant C is. So…

If we say that at **x=a** the area **A** underneath the curve is **0**, watch what happens… Look at what we get…

Which means that:

Hence, we can conclude that:

And this formula can be transformed into something more fancy if we are measuring an area underneath a curve from **x=a** to **x=b**…

This is probably the formula you’re most familiar with…

Which is the formula which can be used to find areas underneath curves.

**If you are still confused and would like to go through this proof once again, please watch my video below…**

You can also leave your comments below.

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