Fundamental theorem of calculus

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This article/section deals with mathematical concepts appropriate for late high school or early college.

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is the remarkable result that the two fundamental operations of calculus are just inverses of each other. Those two operations are performed on functions from the real numbers to the real numbers, and are most easily visualized when the functions are expressed in terms of graphs. The operations are:

• Differentiation—find the slope of a function's graph at a given point.
• Integration—find the area under a graph between two given limits.

The Theorem

There are two parts to the Fundamental Theorem of Calculus

Part 1

The first can be written as: Let the function $f$ be continuous function defined on a closed interval $[a,b]$. Define $F(x)$ as:

$F(x)=\int _{a}^{x}f(t)dt$

It follows that:

The first part states that if a function $F$ is the antiderivative of a function $f$, then the derivative of $F$ is $f$. In other words, antiderivative and derivative are opposite functions.

Part 2

If:

$f(x)=F'(x)$

Then:

$\int _{a}^{b}f(x)dx=F(b)-F(a)$

The second part begins with what we know from part 1.
It then states that the definite integral of the function $f$ from $a$ to $b$ is equal to $F$ evaluated at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} minus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} evaluated at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} .

External links

• Proof for the Fundamental Theorem of Calculus

See also

• Calculus
• Indefinite integral
• Derivative
• Integral