I have thought of an intuitive way to view the link between Area (under the curve) and anti derivatives.
Firstly let's define what area (under the curve) actually is. It's the cumulation of every value taken by a given function multiplied by an infinitesimal value (delta x). these can be represented by infinite rectangles with widths having infinitesimal values blah blah blah whatever...
but what's important is that for a function to cumulate the values of another function, its rate of change has to be equal to the values of the other function.
Let's declare a function "f(x)"
and a function describing its area "A(x)".
We can state according to our previous statement that:
dA/dx= f(x)
dA= f(x)dx
∫dA= ∫f(x)dx
A(x) = ∫f(x)dx
the link between the anti derivative and the Area is now obvious
Firstly let's define what area (under the curve) actually is. It's the cumulation of every value taken by a given function multiplied by an infinitesimal value (delta x). these can be represented by infinite rectangles with widths having infinitesimal values blah blah blah whatever...
but what's important is that for a function to cumulate the values of another function, its rate of change has to be equal to the values of the other function.
Let's declare a function "f(x)"
and a function describing its area "A(x)".
We can state according to our previous statement that:
dA/dx= f(x)
dA= f(x)dx
∫dA= ∫f(x)dx
A(x) = ∫f(x)dx
the link between the anti derivative and the Area is now obvious
Comments
unrelated to the post: