a6314b…: “As well as the wonderful rhombus”

$\newenvironment {prompt}{}{} \newcommand {\ungraded }[0]{} \newcommand {\todo }[0]{} \newcommand {\npnoround }[0]{\nprounddigits {-1}} \newcommand {\npnoroundexp }[0]{\nproundexpdigits {-1}} \newcommand {\npunitcommand }[1]{\ensuremath {\mathrm {#1}}} \newcommand {\RR }[0]{\mathbb R} \newcommand {\R }[0]{\mathbb R} \newcommand {\N }[0]{\mathbb N} \newcommand {\Z }[0]{\mathbb Z} \newcommand {\sagemath }[0]{\textsf {SageMath}} \DeclareMathOperator {\dx }{dx} \DeclareMathOperator {\dt }{dt} \DeclareMathOperator {\dy }{dy} \DeclareMathOperator {\dz }{dz} \DeclareMathOperator {\dr }{dr} \DeclareMathOperator {\dw }{dw} \DeclareMathOperator {\du }{du} \DeclareMathOperator {\dv }{dv} \DeclareMathOperator {\ds }{ds} \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 {\scal }{scal} \DeclareMathOperator {\sign }{sign} \newcommand {\point }[1]{\left (#1\right )} \newcommand {\pt }[1]{\mathbf {#1}} \newcommand {\Lim }[2]{\lim _{\point {#1} \to \point {#2}}} \newcommand {\bar }[0]{\overline } \newcommand {\sectionOutcomes }[0]{} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument }$

Worked Out Examples Problem Videos

The following video may be helpful when trying to solve this example. Note that you may skip to the end of the video to get completion credit for this page if you don’t need to watch it.

Practice Problems: Definite Integrals

1 :

Use limits to find a perfect approximation for the area under the curve of the function $f(x) = \sage {p1f1}$ from $x=\sage {p1start}$ to $x=\sage {p1end}$ . $\answer {\sage {p1ans}}$ .

Remember you can factor out any constants from the function all the way outside of the sum, which will help with the limits. You want to make sure to break the sum apart as much as you can and use the handy identities:

(a): $\sum \limits _{k=1}^N 1 = N$
(b): $\sum \limits _{k=1}^N k = \frac {N(N+1)}{2}$
(c): $\sum \limits _{k=1}^N N^2 = \frac {N(N+1)(2N+1)}{6}$ .

This will help you get rid of the $\Sigma$ part and get to a “closed form” where you can actually evaluate the limit!