Home page for accesible maths Math 101 Chapter 5: Integration 5.6 Direct calculation of areas 5.8 Remarks on the fundamental theorem of calculus

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5.7 The fundamental theorem of calculus

For calculating areas, calculus provides an alternative approach which is often useful. The converse implication is due to Barrow.

Let $f$ be a continuous function so $\int_{a}^{x}f(t)\,dt$ is the area under the graph of $f$ between $a$ and $x$ . Then the function $F(x)=\int_{a}^{x}f(t)\,dt+c$ with $c$ constant is called an indefinite integral of $f$ .

Fundamental Theorem of Calculus

Let $F(x)$ be an indefinite integral of $f$ . Then $F$ has derivative $F^{\prime}(x)=f(x).$

Conversely, if $F$ is any differentiable function such that $F^{\prime}(x)=f(x)$ is continuous for all $x$ , then

\int_{a}^{b}f(t)\,dt=F(b)-F(a).