9E: Chapter Review Exercises
- Page ID
- 26048
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
Exercise \(\PageIndex{1}\)
Sketch the regions and evaluate the integrals:
1) \(\int_1^{\ln 8} \int_0^{\ln y} 2e^{x+y} \, dy dx\)
2) \(\int_0^{2} \int_x^{2} 3y^2 sin({xy}) \, dy dx\)
3) \(\int_0^{1} \int_0^{3} \frac{4x^2}{(y-1)^{2/3}} \, dy dx\)
4) \(\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{3}{(x^2+1)(y^2+1)} \, dy dx\)
5) \(\int \int_{x^2+y^2 \leq 1} \ln (x^2+y^2)\, dA\)
6) \(\int_0^2 \int_{y/2}^1 y e^{x^3} \,dx dy\)
7) \(\int \int_Q \frac{dA}{(1+x^2)(1+y^2)}\), where \(Q\) is the first quadrant of the \(xy-\)plane.
8) \(\int \int_R x \cos (y) \,dA\), where \(R\) is the region bounded by the coordinate axes and the curve \(y=1-x^2.\)
- Answer
-
Add texts here. Do not delete this text first.