Calculus

Fundamentals

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The Fundamental Theorem of Calculus provides a powerful connection between the concepts of derivatives and integrals, making it a cornerstone in the study of calculus. It enables the calculation of definite integrals by finding antiderivatives and vice versa, offering a more efficient way to compute certain types of integrals.

First Fundamental Theorem

Let $f$ be a continuous real-valued function defined on a closed interval $[a, b]$, and let $F$ be the function defined by:

$$F(x)= \int_{a}^{x}f(t) dt.$$

Then, $F$ is continuous on $[a, b]$, differentiable on the open interval $[a, b]$, and $F'(x) = f(x)$ for all $x$ in $(a, b)$.

In simpler terms, if $f$ is a continuous function, and $F$ is the accumulation (or antiderivative) of $f$, then the derivative of $F$ is $f$.

Second Fundamental Theorem

Let $f$ be a real-valued function defined on a closed interval $[a, b]$, and let $F$ be any antiderivative of $f$ on $[a, b]$. Then,

$$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$$

In other words, the definite integral of a function $f$ over an interval $[a, b]$ is equal to the difference between the antiderivative $F$ evaluated at ( b ) and $F$ evaluated at $a$.

tip

Let $f:[a,b] \to \R$ be Riemann integrable. Let $F:[a,b]\to\R$ be $F(x)= \int_{a}^{x}f(t)dt$. Then $F$ is continuous, and at all $x$ such that $f$ is continuous at $x$, $F$ is differentiable at $x$ with $F'(x)=f(x)$.