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
yi is going to be the length and Δxi 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
Atotal= y1Δx1+y2Δx2…+yn-1Δxn-1+ynΔxn 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 En 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.
scalculus 