The Fundamental Theorem of Calculus

The fundamental theorem of calculus is a critical portion of calculus because it links the concept of a derivative to that of an integral. As a result, we can use our knowledge of derivatives to find the area under the curve, which is often quicker and simpler than using the definition of the integral.

Mean Value Theorem for Integration

We will need the following theorem in the discussion of the Fundamental Theorem of Calculus.

Suppose $f(x)$ is continuous on $[a,b]$ . Then ${\frac {1}{b-a}}\int \limits _{a}^{b}f(x)dx=f(c)$ for some $c\in [a,b]$.

Proof of the Mean Value Theorem for Integration

$f(x)$ satisfies the requirements of the Extreme Value Theorem, so it has a minimum $m$ and a maximum $M$ in $[a,b]$. Since

$\int \limits _{a}^{b}f(x)dx=\lim _{n\to \infty }\sum _{k=1}^{n}f(x_{k}^{*})\cdot {\frac {b-a}{n}}=\lim _{n\to \infty }{\frac {b-a}{n}}\cdot \sum _{k=1}^{n}f(x_{k}^{*})$

and since

$m\leq f(x_{k}^{*})\leq M$ for all $x_{k}^{*}\in [a,b]$

we have

${\begin{aligned}&\lim _{n\to \infty }{\frac {b-a}{n}}\cdot \sum _{k=1}^{n}m\leq \lim _{n\to \infty }{\frac {b-a}{n}}\cdot \sum _{k=1}^{n}f(x_{k}^{*})\leq \lim _{n\to \infty }{\frac {b-a}{n}}\cdot \sum _{k=1}^{n}M\\&\lim _{n\to \infty }mn\cdot {\frac {b-a}{n}}\leq \int \limits _{a}^{b}f(x)dx\leq \lim _{n\to \infty }Mn\cdot {\frac {b-a}{n}}\\&\lim _{n\to \infty }m(b-a)\leq \int \limits _{a}^{b}f(x)dx\leq \lim _{n\to \infty }M(b-a)\\&m(b-a)\leq \int \limits _{a}^{b}f(x)dx\leq M(b-a)\\&m\leq {\frac {1}{b-a}}\int \limits _{a}^{b}f(x)dx\leq M\end{aligned}}$

Since $f$ is continuous, by the Intermediate Value Theorem there is some $f(c)$ with $c\in [a,b]$ such that

${\frac {1}{b-a}}\int \limits _{a}^{b}f(x)dx=f(c)$

Fundamental Theorem of Calculus

Statement of the Fundamental Theorem

Suppose that $f$ is continuous on 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,b]} . We can define a function $F$ by

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(x)=\int\limits_a^x f(t)dt\quad\text{for }x\in[a,b]}

When we have such functions 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} and $f$ where 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'(x)=f(x)} for every 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 x} in some interval 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 I} we say that 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} is the antiderivative of 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} on 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 I} .

Figure 1

Note: a minority of mathematicians refer to part one as two and part two as one. All mathematicians refer to what is stated here as part 2 as The Fundamental Theorem of Calculus.

Proofs

Proof of Fundamental Theorem of Calculus Part I

Suppose $x\in (a,b)$ . Pick 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 \Delta x} so that 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 x+\Delta x\in(a,b)} . Then

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

and

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(x+\Delta x)=\int\limits_a^{x+\Delta x}f(t)dt}

Subtracting the two equations gives

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(x+\Delta x)-F(x)=\int\limits_a^{x+\Delta x}f(t)dt-\int\limits_a^x f(t)dt}

Now

$\int \limits _{a}^{x+\Delta x}f(t)dt=\int \limits _{a}^{x}f(t)dt+\int \limits _{x}^{x+\Delta x}f(t)dt$

so rearranging this we have

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(x+\Delta x)-F(x)=\int\limits_x^{x+\Delta x}f(t)dt}

According to the Mean Value Theorem for Integration, there exists a 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 c\in[x,x+\Delta x]} such that

$\int \limits _{x}^{x+\Delta x}f(t)dt=f(c)\cdot \Delta x$

Notice that 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 c} depends on 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 \Delta x} . Anyway what we have shown is that

$F(x+\Delta x)-F(x)=f(c)\cdot \Delta x$

and dividing both sides by 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 \Delta x} gives

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 \frac{F(x+\Delta x)-F(x)}{\Delta x}=f(c)}

Take the limit as $\Delta x\to 0$ we get the definition of the derivative of 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} at $x$ so we have

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'(x)=\lim_{\Delta x\to0}\frac{F(x+\Delta x)-F(x)}{\Delta x}=\lim_{\Delta x\to0}f(c)}

To find the other limit, we will use the squeeze theorem. 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 c\in[x,x+\Delta x]} , so $x\leq c\leq x+\Delta x$ . Hence,

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 \lim_{\Delta x\to0}\Big[x+\Delta x\Big]=x\quad\Rightarrow\quad\lim_{\Delta x\to0}c=x}

As 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} is continuous we have

$F'(x)=\lim _{\Delta x\to 0}f(c)=f\left(\lim _{\Delta x\to 0}c\right)=f(x)$

which completes the proof.

$\blacksquare$

Proof of Fundamental Theorem of Calculus Part II

Define $P(x)=\int \limits _{a}^{x}f(t)dt$ . Then by the Fundamental Theorem of Calculus part I we know that $P$ is differentiable on $(a,b)$ and for all $x\in (a,b)$

$P'(x)=f(x)$

So $P$ is an antiderivative of $f$ . Since we were assuming that $F$ was also an antiderivative for all $x\in (a,b)$ ,

${\begin{aligned}&P'(x)=F'(x)\\&P'(x)-F'(x)=0\\&{\Big (}P(x)-F(x){\Big )}'=0\end{aligned}}$

Let $g(x)=P(x)-F(x)$ . The Mean Value Theorem applied to $g(x)$ on $[a,\xi ]$ with $a<\xi <b$ says that

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 \frac{g(\xi)-g(a)}{\xi-a}=g'(c)}

for some 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 c} in $(a,\xi )$ . But since 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 g'(x)=0} for all 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 x} in $[a,b]$ , 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 g(\xi)} must equal 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 g(a)} for all 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 \xi} in 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,b)} , i.e. g(x) is constant on $(a,b)$ .

This implies there is a constant 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 C=g(a)=P(a)-F(a)=-F(a)} such that for all 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 x\in(a,b)} ,

$P(x)=F(x)+C$

and as 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 g} is continuous we see this holds when $x=a$ and 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 x=b} as well. And putting 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 x=b} gives

$\int \limits _{a}^{b}f(t)dx=P(b)=F(b)+C=F(b)-F(a)$

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 \blacksquare}

Notation for Evaluating Definite Integrals

The second part of the Fundamental Theorem of Calculus gives us a way to calculate definite integrals. Just find an antiderivative of the integrand, and subtract the value of the antiderivative at the lower bound from the value of the antiderivative at the upper bound. That is

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 \int\limits_a^b f(x)dx=F(b)-F(a)}

where $F'(x)=f(x)$ . As a convenience, we use the notation

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(x)\bigg|_a^b}

to represent 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(b)-F(a)}

Integration of Polynomials

Using the power rule for differentiation we can find a formula for the integral of a power using the Fundamental Theorem of Calculus. Let $f(x)=x^{n}$ . We want to find an antiderivative for 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} . Since the differentiation rule for powers lowers the power by 1 we have that

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 \frac{d}{dx}x^{n+1}=(n+1)x^n}

As long as 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 n+1\ne0} we can divide by 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 n+1} to get

${\frac {d}{dx}}\left({\frac {x^{n+1}}{n+1}}\right)=x^{n}=f(x)$

So the function 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(x)=\frac{x^{n+1}}{n+1}} is an antiderivative of 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} . If $0\notin [a,b]$ then 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} is continuous on $[a,b]$ and, by applying the Fundamental Theorem of Calculus, we can calculate the integral of $f$ to get the following rule.

Notice that we allow all values of $n$ , even negative or fractional. If $n>0$ then this works even if $[a,b]$ includes $0$ .

Examples

• To find $\int \limits _{1}^{2}x^{3}dx$ we raise the power by 1 and have to divide by 4. So

$\int \limits _{1}^{2}x^{3}dx={\frac {x^{4}}{4}}{\Bigg |}_{1}^{2}={\frac {2^{4}}{4}}-{\frac {1^{4}}{4}}={\frac {15}{4}}$

• The power rule also works for negative powers. For instance

$\int \limits _{1}^{3}{\frac {dx}{x^{3}}}=\int \limits _{1}^{3}x^{-3}dx={\frac {x^{-2}}{-2}}{\Bigg |}_{1}^{3}={\frac {1}{-2}}\left(3^{-2}-1^{-2}\right)=-{\frac {1}{2}}\left({\frac {1}{3^{2}}}-1\right)=-{\frac {1}{2}}\left({\frac {1}{9}}-1\right)={\frac {1}{2}}\cdot {\frac {8}{9}}={\frac {4}{9}}$

• We can also use the power rule for fractional powers. For instance

$\int \limits _{0}^{5}{\sqrt {x}}dx=\int \limits _{0}^{5}x^{\frac {1}{2}}dx={\frac {x{\sqrt {x}}}{\frac {3}{2}}}{\Bigg |}_{0}^{5}={\frac {2}{3}}\left(5^{\frac {3}{2}}-0^{\frac {3}{2}}\right)={\frac {10{\sqrt {5}}}{3}}$

• Using linearity the power rule can also be thought of as applying to constants. For example,

$=\int \limits _{3}^{11}7dx=\int \limits _{3}^{11}7x^{0}dx=7\int \limits _{3}^{11}x^{0}dx=7x{\Bigg |}_{3}^{11}=7(11-3)=56$

• Using the linearity rule we can now integrate any polynomial. For example

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 \int\limits_0^3(3x^2+4x+2)dx=(x^3+2x^2+2x)\Bigg|_0^3=3^3+2\cdot 3^2+2\cdot3-0=27+18+6=51}

Exercises

The Fundamental Theorem of Calculus, Part 1

• Definite Integral & Antiderivatives (Slides 6&7). PowerPoint file created by Professor Cynthia Roberts, UTSA.
• Fundamental Theorem of Calculus Part 1. PowerPoint file created by Professor Cynthia Roberts, UTSA.
• The Fundamental Theorem of Calculus PowerPoint file created by Dr. Sara Shirinkam, UTSA.

• Fundamental Theorem of Calculus Part 1 by patrickJMT
• PART 1 OF THE DREADED FUNDAMENTAL THEOREM OF CALCULUS! by Krista King
• Fundamental Theorem of Calculus Part 1 by The Organic Chemistry Tutor
• The Second Fundamental Theorem of Calculus by James Sousa, Math is Power 4U
• Ex 1: The Second Fundamental Theorem of Calculus by James Sousa, Math is Power 4U
• Ex 2: The Second Fundamental Theorem of Calculus (Reverse Order) by James Sousa, Math is Power 4U
• Ex 3: The Second Fundamental Theorem of Calculus by James Sousa, Math is Power 4U
• Ex 4: The Second Fundamental Theorem of Calculus with Chain Rule by James Sousa, Math is Power 4U
• Ex 5: The Second Fundamental Theorem of Calculus with Chain Rule by James Sousa, Math is Power 4U
• Ex 6: Second Fundamental Theorem of Calculus with Chain Rule by James Sousa, Math is Power 4U
• Ex 7: Second Fundamental Theorem of Calculus with Chain Rule by James Sousa, Math is Power 4U

The Fundamental Theorem of Calculus, Part 2

• The Fundamental Theorem of Calculus by James Sousa, Math is Power 4U
• The Fundamental Theorem of Calculus Part 2 by patrickJMT
• PART 2 OF THE FUNDAMENTAL THEOREM OF CALCULUS! by Krista King
• The Fundamental Theorem of Calculus Part 2 by The Organic Chemistry Tutor