Skip to main content
Contents Index Calc Dark Mode Prev Up Next \(\newcommand{\R}{\mathbb{R}}
\newcommand{\va}{\vec{a}}
\newcommand{\vb}{\vec{b}}
\newcommand{\vc}{\vec{c}}
\newcommand{\vC}{\vec{C}}
\newcommand{\vd}{\vec{d}}
\newcommand{\ve}{\vec{e}}
\newcommand{\cursedihat}{\hat{\dot{i}}}
\newcommand{\vi}{\hat{\imath}}
\newcommand{\vj}{\hat{\jmath}}
\newcommand{\vk}{\hat{k}}
\newcommand{\vn}{\vec{n}}
\newcommand{\vm}{\vec{m}}
\newcommand{\vr}{\vec{r}}
\newcommand{\vs}{\vec{s}}
\newcommand{\vu}{\vec{u}}
\newcommand{\vv}{\vec{v}}
\newcommand{\vw}{\vec{w}}
\newcommand{\vx}{\vec{x}}
\newcommand{\vy}{\vec{y}}
\newcommand{\vz}{\vec{z}}
\newcommand{\vzero}{\vec{0}}
\newcommand{\vF}{\vec{F}}
\newcommand{\vG}{\vec{G}}
\newcommand{\vH}{\vec{H}}
\newcommand{\vR}{\vec{R}}
\newcommand{\vT}{\vec{T}}
\newcommand{\vN}{\vec{N}}
\newcommand{\vL}{\vec{L}}
\newcommand{\vB}{\vec{B}}
\newcommand{\vS}{\vec{S}}
\newcommand{\proj}{\text{proj}}
\newcommand{\comp}{\text{comp}}
\newcommand{\nin}{}
\newcommand{\vecmag}[1]{\left\lVert #1\right\rVert}
\newcommand{\grad}{\nabla}
\newcommand\restrict[1]{\raise-.5ex\hbox{$\Big|$}_{#1}}
\DeclareMathOperator{\curl}{curl}
\DeclareMathOperator{\divg}{div}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Activity 12.1.4 .
Let
\(f(x,y) = \sqrt{4-y^2}\) on the rectangular domain
\(R = [1,7] \times [-2,2]\) where we partition
\([1,7]\) into 3 equal length subintervals and
\([-2,2]\) into 2 equal length subintervals. A table of values of
\(f\) at some points in
\(R\) is given in
Table 12.1.14 , and a graph of
\(f\) with the indicated partitions is shown in
Figure 12.1.15 .
Table 12.1.14. Table of values of \(f(x,y) = \sqrt{4-y^2}\text{.}\)
\(x \downarrow \backslash \, y \rightarrow\) \(-2\) \(-1\) \(0\) \(1\) \(2\)
\(1\) \(0\) \(\sqrt{3}\) \(2\) \(\sqrt{3}\) \(0\)
\(2\) \(0\) \(\sqrt{3}\) \(2\) \(\sqrt{3}\) \(0\)
\(3\) \(0\) \(\sqrt{3}\) \(2\) \(\sqrt{3}\) \(0\)
\(4\) \(0\) \(\sqrt{3}\) \(2\) \(\sqrt{3}\) \(0\)
\(5\) \(0\) \(\sqrt{3}\) \(2\) \(\sqrt{3}\) \(0\)
\(6\) \(0\) \(\sqrt{3}\) \(2\) \(\sqrt{3}\) \(0\)
\(7\) \(0\) \(\sqrt{3}\) \(2\) \(\sqrt{3}\) \(0\)
Figure 12.1.15.
(a) Sketch the region
\(R\) in the plane using the values in
Table 12.1.14 as the partitions.
(b) Calculate the double Riemann sum using the given partition of
\(R\) and the values of
\(f\) in the upper right corner of each subrectangle.
(c) Use geometry to calculate the exact value of
\(\iint_R f(x,y) \, dA\) and compare it to your approximation. Describe one way we could obtain a better approximation using the given data.
