area

As we all know the formula for a rectangular area is : area=length•width

In integral calculus, the same formula is used but there are some substitution:

example:

fig.1 AREA

y_i is going to be the length and Δx_i is going to be the width.

so, to substitute, we have:

area=l•w

l=y₁ 

Δx₁ =x₂-x₁  w=Δx₁

area=y₁Δx₁

But of course,  we are not going to compute on one area all the time. Sometimes we are going to encounter  figures that require multiple area computation. See in fig.2.

fig.2 AREA

A_total= y₁Δx₁+y₂Δx₂…+y_n-1Δx_n-1+y_nΔx_{n         or      the summation of all areas, in formulas we have:}  

area formula derivation

As we can see in fig.2, y=f(x)

we substitute f(x) to y:

area formula derivation2

You must be thinking what E_n  is. E stands for error. See fig.1 and fig.2. Error is added to the equation because the rectangular area we are solving has a curve with it. See fig.1 and fig.2.

To remove error in the equation, we set the limits. Limits are the boundaries of what we solving. Since we don’t know yet the limit (and for formula purposes also) we set n to infinity (∞).

Then, we change the limit to ∫x=b x=a, and Δx to dx.

And the final formula:

area final formula

Example in fig. 2, we want to solve the area from x1 to x2. So we set the limit as b=x2 and a=x1. The value of b is always greater than a.