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\begin{document}

\title{MTH 1420, Spring 2012\\ Dr. Graham-Squire}
\author{Section 5.3:  Definite Integrals}
\date{}
\maketitle

HW: 4, 11, 18, 25, 31, 44

Practice: 5, 8, 15, 19, 23, 39, 42 (no need to graph), 46\endnote{Answers for evens: \#8: 10e, \#42: $e^x-(2/3)x^3+c$, \#46:$\tan t +\sec t+ C$}, 53  

\section{Introduction}

In the previous sections we talked about how to approximate the area under a curve, and we also discussed a method of using a limit to calculate the \emph{actual} area under a curve (as opposed to just an approximation).  In this section we give a much faster method for calculating the area under a curve, which is known as integration.


\section{Indefinite Integrals} 

In Calculus 1 we discuss the operation of taking an \emph{antiderivative} of a function.  We now present a notation for this.

\bdef
	We say that a function $F(x)$ is an \textbf{indefinite integral} of a function $f(x)$ if $F'(x)=f(x)$.  We write 
	\[F(x)=\int f(x)dx
	\]
	The ``dx" is a notational place holder and does not have any bearing on the calculation. Note that for a given function $f$ there are an infinite number of indefinite integrals (that is, antiderivatives). 
\edefi

\bexa
	Calculate the general indefinite integral $\int (x^3 +3x^2)dx$.
\eexa
\newpage

\bexe
	Calculate the general indefinite integrals for the following:
	\benu[(a)]
		\item $\int \sin x \,dx$
		\vspace{.5in}
		\item $\d\int \sqrt[4]{x^7}dx$
		\vspace{.5in}
		\item $\int \csc x\cot x \,dx$
		\vspace{.5in}
		\item $\d\int e^x+ \frac{-1}{\sqrt{1-x^2}}dx$
		\vspace{.5in}
		\item $\d \int (\ln 6) 6^x \,dx$
		\vspace{.5in}
		\item $\d\int \frac{x^7+x^3+1}{x^4}dx$
		\vspace{.5in}
		
		These last two are tricky:
		
		\item $\int -\csc^2(10x) dx$
		\vspace{.7in}
		\item  $\int (5x^4\sin x+x^5\cos x)dx$
		\vspace{.9in}
	\eenu
\eexe


\section{Definite Integrals}

A \emph{definite integral} is similar to an indefinite integral notation-wise (it just has limits of integration). This is because indefinite integrals can be used to measure the area under a curve.

\bdef
	If $f$ is continuous on an interval $[a,b]$, the definite integral of $f$ from $a$ to $b$ is 
	\[ \int^b_a f(x)dx=F(b)-F(a)
	\]
	where $F$ is \emph{any} indefinite integral (aka antiderivative) of $f$ (that is, any $F$ such that $F'=f$).
\edefi

\bexa
	Calculate the definite integral $\d\int_2^4 (x^3 +3x^2)dx$.  Use two different antiderivatives to see that you get the same answer.
\eexa
\vspace{2.5in}

\bexe
	Calculate the definite integral $\d\int_0^2 x^3\,dx$.  Compare this answer to Examples 3 and 4 from Section 5.1 when you are finished.
\eexe
\vspace{2in}

\bthe
	The definite integral $ \int^b_a f(x)dx$ measures the area between the curve $f$ and the $x$-axis in the interval $[a,b]$, as long as that area is \underline{positive}.  If $f$ is below the $x$-axis then the definite integral will give you the negative of the area, and if $f$ crosses the $x$-axis in the interval then the definite integral gives you the difference between the areas above and below the $x$-axis (that is, the positive area above plus the negative area below).
\ethe
\newpage

An example that illustrates the difference between the definite integral and the area under a curve is as follows:

\bexa
	The velocity function for a particle on a line is $v(t)=t^2-2t-8$ meters/second, for $1\leq t\leq 6$.  Find (a) the displacement and (b) the total distance traveled.
\eexa
\vspace{4in}

\bexa
	Calculate the definite integral $\d\int_0^{\pi/3}\frac{\sin \theta +\sin \theta \tan^2\theta}{\sec^2\theta}d\theta$.
\eexa
\newpage

\bexe
	Evaluate $\d\int^8_1\sqrt[3]{x}\,dx$.
\eexe
\vspace{3in}


\bexe
	Evaluate $\d\int_0^{\pi} (e^x-\sin x)dx$.
\eexe
\newpage

\bexe
	Consider the curve $f(x)=x^2(x-3)$.  Calculate (a) the definite integral for $f$ from 0 to 4, as well as (b) the area between $f$ and the $x$-axis from 0 to 4.  Note: you should get a different answer for (a) and (b).
\eexe
\vspace{5in}

\bexe
	A honeybee population starts with 100 bees and increases at a rate of $n'(t)$ bees per week.  What does \[100+\int^{15}_0 n'(t)dt \] represent?
\eexe
\vspace{1in}
\theendnotes
\end{document} 