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\parbox{2.25in}{\bf Math 3220}
\hfill {\Large \bf Homework \#8} \hfill
\parbox{2.25in}{\bf \hfill Due: Mon., Apr. $29^\text{th}$, 2019}

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%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
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\begin{description}
\item[\#1 Riemann Sums] It might be helpful to know that: $\sum\limits_{k=1}^n 1 = n$, \quad $\sum\limits_{k=1}^n k = \dfrac{n(n+1)}{2}$, \quad or \quad $\sum\limits_{k=1}^n k^2 = \dfrac{n(n+1)(2n+1)}{6}$.

\begin{enumerate}[(a)]
\item Consider $I=\int_{-1}^2 x^2\,dx$. Find the regular partition of $[-1,2]$ into $n=6$ subintervals. 

Compute Riemann sums using (i) left, (ii), right, and (iii) midpoint sample points.
\item Briefly explain why $f(x)=x^2$ is integrable on $[-1,2]$. Find a general formula for a right hand (Riemann) sum approximation of $I=\int_{-1}^2 x^2\,dx$ when using a regular partition of $[-1,2]$ into $n$ subintervals. 
\item Compute your sum from part (b), then use that to compute $I$. 

Confirm your result by using the Fundamental Theorem of Calculus.
\end{enumerate}

\item[\#2 Integral Estimating] Suppose that $f$ is continuous on $[0,2]$, $f(0)=5$, and $f'(x)>0$ for all $0<x<2$. 

Show that $\int_0^2 f \geq 10$.

\item[\#3 Squeezing] Let $g$ be integrable on $[a,b]$ with $\int_a^b g=0$. Suppose that $0 \leq f(x) \leq g(x)$ for all $x \in [a,b]$. 

Show that $f$ is integrable on $[a,b]$ and that $\int_a^b f =0$.

\item[\#4 Linear] Let $f$ and $g$ be integrable on $[a,b]$. 

Use the definition of the Riemann integral to show that $f+g$ is integrable with $\int_a^b (f+g) = \int_a^b f + \int_a ^b g$.

\item[RESUBMIT] Type up Homework \#7 Problem \#2 and its solution in \LaTeX. 

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