Why Integration Gives The Area (The Fundamental Theorem of Calculus))

The First Fundamental Theorem of Calculus shows why integrals give the area under a curve. Consider a function, \(f(x)\) and the function which outputs the area under \(f(x)\), \(A(x)\):


The area up to \(x+h\) (i.e. A(x+h)) is approximately the area up to x (i.e. \(A(x)\)) plus the area of the rectangular strip:

\[A(x+h) \approx A(x)+(f(x) \times h)\]

We can rearrange to give: \[f(x) \approx \frac{A(x+h)-A(x)}{h}\]

As \(h \to 0\), the rectangle becomes infinitely thin and this becomes an equality:

\[f(x)=\lim_{h\to 0} \frac{A(x+h)-A(x)}{h}\]

However, we recognise this to be the formula for differentiating from first principles, which means that:

\[\lim_{h\to 0} \frac{A(x+h)-A(x)}{h}=A'(x)\]

This gives us that \(f(x)=A'(x)\).


Integrating both sides with respect to \(x\) between \(a\) and \(b\) gives:

\[\int_{a}^{b} f(x) dx = \int_{a}^{b} A'(x) dx\]

\[\therefore \int_{a}^{b} f(x) dx = A(b)-A(a)\]

Which is the area between \(a\) and \(b\), and so we have proven that the definite integral gives the area under the graph between two points.


This final bit is called The Second Fundamental Theorem of Calculus (also known as the Newton-Leibniz axiom).

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