The Second Fundamental Theorem of Calculus

$\newenvironment {prompt}{}{} \newcommand {\todo }[0]{} \newcommand {\npnoround }[0]{\nprounddigits {-1}} \newcommand {\npnoroundexp }[0]{\nproundexpdigits {-1}} \newcommand {\npunitcommand }[1]{\ensuremath {\mathrm {#1}}} \newcommand {\CancelColor }[0]{} \newcommand {\cancelto }[0]{1} \newcommand {\RR }[0]{\mathbb R} \newcommand {\R }[0]{\mathbb R} \newcommand {\N }[0]{\mathbb N} \newcommand {\Z }[0]{\mathbb Z} \newcommand {\d }[0]{\mathop {}\!d} \newcommand {\l }[0]{\ell } \newcommand {\ddx }[0]{\frac {d}{\d x}} \newcommand {\zeroOverZero }[0]{\ensuremath {\boldsymbol {\tfrac {0}{0}}}} \newcommand {\inftyOverInfty }[0]{\ensuremath {\boldsymbol {\tfrac {\infty }{\infty }}}} \newcommand {\zeroOverInfty }[0]{\ensuremath {\boldsymbol {\tfrac {0}{\infty }}}} \newcommand {\zeroTimesInfty }[0]{\ensuremath {\small \boldsymbol {0\cdot \infty }}} \newcommand {\inftyMinusInfty }[0]{\ensuremath {\small \boldsymbol {\infty -\infty }}} \newcommand {\oneToInfty }[0]{\ensuremath {\boldsymbol {1^\infty }}} \newcommand {\zeroToZero }[0]{\ensuremath {\boldsymbol {0^0}}} \newcommand {\inftyToZero }[0]{\ensuremath {\boldsymbol {\infty ^0}}} \newcommand {\numOverZero }[0]{\ensuremath {\boldsymbol {\tfrac {\#}{0}}}} \newcommand {\dfn }[0]{\textbf } \newcommand {\unit }[0]{\mathop {}\!\mathrm } \newcommand {\eval }[1]{\bigg [ #1 \bigg ]} \newcommand {\seq }[1]{\left ( #1 \right )} \newcommand {\epsilon }[0]{\varepsilon } \newcommand {\phi }[0]{\varphi } \newcommand {\iff }[0]{\Leftrightarrow } \DeclareMathOperator {\arccot }{arccot} \DeclareMathOperator {\arcsec }{arcsec} \DeclareMathOperator {\arccsc }{arccsc} \DeclareMathOperator {\si }{Si} \DeclareMathOperator {\proj }{\vec {proj}} \DeclareMathOperator {\scal }{scal} \DeclareMathOperator {\sign }{sign} \newcommand {\arrowvec }[0]{\overrightarrow } \newcommand {\vec }[0]{\mathbf } \newcommand {\veci }[0]{{\boldsymbol {\hat {\imath }}}} \newcommand {\vecj }[0]{{\boldsymbol {\hat {\jmath }}}} \newcommand {\veck }[0]{{\boldsymbol {\hat {k}}}} \newcommand {\vecl }[0]{\boldsymbol {\l }} \newcommand {\uvec }[1]{\mathbf {\hat {#1}}} \newcommand {\utan }[0]{\mathbf {\hat {t}}} \newcommand {\unormal }[0]{\mathbf {\hat {n}}} \newcommand {\ubinormal }[0]{\mathbf {\hat {b}}} \newcommand {\dotp }[0]{\bullet } \newcommand {\cross }[0]{\boldsymbol \times } \newcommand {\grad }[0]{\boldsymbol \nabla } \newcommand {\divergence }[0]{\grad \dotp } \newcommand {\curl }[0]{\grad \cross } \newcommand {\lto }[0]{\mathop {\longrightarrow \,}\limits } \newcommand {\bar }[0]{\overline } \newcommand {\surfaceColor }[0]{violet} \newcommand {\surfaceColorTwo }[0]{redyellow} \newcommand {\sliceColor }[0]{greenyellow} \newcommand {\vector }[1]{\left \langle #1\right \rangle } \newcommand {\sectionOutcomes }[0]{} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument } \newcommand {\epstopdfsetup }[0]{\setkeys {ETE}} changed the theorem header of amsthm failed\MessageBreak }}} \newcommand {\GetTitleStringSetup }[0]{\setkeys {gettitlestring}} \newcommand {\Sectionformat }[2]{#1}$ $% This file was *autogenerated* from digInSecondFundamentalTheoremOfCalculus.sagetex.sage with % sagetex.py version 2015/08/26 v3.0-92d9f7a %69636c186a21514eacdef816933edd89% md5sum of corresponding .sage file (minus "goboom","current_tex_line",and pause/unpause lines)$

The accumulation of a rate is given by the change in the amount.

There is a another common form of the Fundamental Theorem of Calculus:

Second Fundamental Theorem of Calculus Let $f$ be continuous on $[a,b]$ . If $F$ is any antiderivative of $f$ , then $\int _a^b f(x)\d x = F(b)-F(a).$

Let $a\le c\le b$ and write $\begin{align*} \int_a^b f(x) \d x &= \int_a^c f(x) \d x + \int_c^b f(x) \d x \\ &= \int_c^b f(x) \d x - \int_c^a f(x) \d x. \end{align*}$

By the First Fundamental Theorem of Calculus, we have $F(b) = \int _c^b f(x) \d x\qquad \text {and}\qquad F(a) = \int _c^a f(x) \d x$ for some antiderivative $F$ of $f$ . So $\int _a^b f(x) \d x = F(b)-F(a)$ for this antiderivative. However, any antiderivative could have be chosen, as antiderivatives of a given function differ only by a constant, and this constant always cancels out of the expression when evaluating $F(b)-F(a)$ .

From this you should see that the two versions of the Fundamental Theorem are very closely related. In reality, the two forms are equivalent, just differently stated. Hence people often simply call them both ‘‘The Fundamental Theorem of Calculus.’’ One way of thinking about the Second Fundamental Theorem of Calculus is:

This could be read as:

The blackblueaccumulation of a greenrate is given by the blueblackchange in the amount.

When we compute a definite integral, we first find an antiderivative and then evaluate at the limits of integration. It is convenient to first display the antiderivative and then evaluate. A special notation is often used in the process of evaluating definite integrals using the Fundamental Theorem of Calculus. Instead of explicitly writing $F(b)-F(a)$ , we often write $\eval {F(x)}_a^b$ meaning that one should evaluate $F(x)$ at $b$ and then subtract $F(x)$ evaluated at $a$ $\eval {F(x)}_a^b = F(b)-F(a).$

Let’s see some examples of the fundamental theorem in action.

Compute: $\int _{-2}^2 x^3\d x$

We start by finding an antiderivative of $x^3$ . A correct choice is $\frac {x^{4}}{4}$ , one could verify this by taking the derivative. Hence $\begin{align*} \int_{-2}^2 x^3 \d x &=\eval{\frac{x^4}{4}}_{-2}^2\\ &= \frac{2^4}{4} - \frac{(-2)^4}{4}\\ &= 0. \end{align*}$

Compute: $\int _0^5 e^t\d t$

We start by finding an antiderivative of $e^t$ . A correct choice is $\answer [given]{e^t}$ , one could verify this by taking the derivative. Hence $\begin{align*} \int_0^5 e^t \d t &= \eval{e^t}_0^5 \\ &= e^5-\answer[given]{e^0}\\ &= e^5-\answer[given]{1}. \end{align*}$

Compute: $\int _1^2\left (x^9 + \frac {1}{x}\right ) \d x$

We start by finding an antiderivative of $x^9 + \frac {1}{x}$ . A correct choice is $\frac {x^{10}}{10} + \ln (x)$ , one could verify this by taking the derivative. Hence $\begin{align*} \int_1^2\left(x^9 + \frac{1}{x}\right) \d x &= \eval{\frac{x^{10}}{10} + \ln(x)}_1^2 \\ &= \frac{2^{10}}{10} + \ln(2) - \answer[given]{\frac{1}{10}}. \end{align*}$

Now we know that to solve certain kinds of problems, those that involve accumulation of some form, we ‘‘merely’’ find an antiderivative and substitute two values and subtract. Unfortunately, finding antiderivatives can be quite difficult. While there are a small number of rules that allow us to compute the derivative of any common function, there are no such rules for antiderivatives. There are some techniques that frequently prove useful, but we will never be able to reduce the problem to a completely mechanical process.