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Calculus 3 : Multiple Integration

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Example Questions

Example Question #601 : Multiple Integration

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(\frac{(25sin(\frac{x^{2}}{2}+\frac{ y^{2}}{2}))}{2}-\frac{ (4cos(x^{2} + y^{2}))}{15})dA\text{, where D is}\\&\text{the region defined between two circles centered on the origin with radii }\\&0.11\text{ and }1.47\\&\text{and encompassed by vectors from the origin }\\&\overrightarrow{u_1}=(0.790,0.613)\text{ and }\overrightarrow{u_2}=(-0.249,-0.969)\\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

Possible Answers:

-148.34

-12.36

6.18

24.72

Correct answer:

24.72

Explanation:

$\begin{align*}&\text{The current form of the integral is rather unwieldy, due to the }x^2\text{ and }y^2\text{ terms.}\\&\text{An approach that would be beneficial is a conversion to polar form:}\\&r=cos(\theta);r=sin(\theta)\\&r^2=x^2+y^2\\&dA=rdrd\theta\\&\text{With this we can find: }\iint_{D}(\frac{(25sin(\frac{x^{2}}{2}+\frac{ y^{2}}{2}))}{2}-\frac{ (4cos(x^{2} + y^{2}))}{15})dA\rightarrow\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}(\frac{(25\cdot rsin(\frac{r^{2}}{2}))}{2}-\frac{ (4\cdot rcos(r^{2}))}{15})drd\theta\\&\text{The bounds of our radius we can readily take from the problem. To find the angle ranges:}\end{align*}$

$\begin{align*}&\theta_1=arctan(\frac{0.613}{0.790})=0.21\pi;\theta_2=arctan(\frac{-0.969}{-0.249})=1.42\pi\\&\text{Now, utilizing integral rules:}\\&\int[rsin(ar^2)]=-\frac{cos(ar^2)}{2a}=-\frac{cos^2(\frac{a}{2}r^2)}{a}\\&\int[rcos(ar^2)]=\frac{sin(ar^2)}{2a}\\&\int_{0.21\pi}^{1.42\pi}\int_{0.11}^{1.47}(\frac{(25\cdot rsin(\frac{r^{2}}{2}))}{2}-\frac{ (4\cdot rcos(r^{2}))}{15})drd\theta=(-\frac{ (2sin(r^{2}))}{15}-\frac{ (25cos(\frac{r^{2}}{2}))}{2})d\theta|_{0.11}^{1.47}\\&\int_{0.21\pi}^{1.42\pi}(6.504)d\theta=(6.504\theta)|_{0.21\pi}^{1.42\pi}=24.72\end{align*}$

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Example Question #118 : Double Integrals

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(-\frac{ (37cos(\frac{(2x^{2})}{3}+\frac{ (2y^{2})}{3}))}{5}- 5sin(x^{2} + y^{2}))dA\text{, where D is}\\&\text{the region defined between two circles centered on the origin with radii }\\&0.36\text{ and }1.33\\&\text{and encompassed by vectors from the origin }\\&\overrightarrow{u_1}=(0.661,0.750)\text{ and }\overrightarrow{u_2}=(-0.998,-0.063)\\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

Possible Answers:

35.92

2.99

-3.59

-17.96

Correct answer:

-17.96

Explanation:

$\begin{align*}&\text{The current form of the integral is rather unwieldy, due to the }x^2\text{ and }y^2\text{ terms.}\\&\text{An approach that would be beneficial is a conversion to polar form:}\\&r=cos(\theta);r=sin(\theta)\\&r^2=x^2+y^2\\&dA=rdrd\theta\\&\text{With this we can find: }\iint_{D}(-\frac{ (37cos(\frac{(2x^{2})}{3}+\frac{ (2y^{2})}{3}))}{5}- 5sin(x^{2} + y^{2}))dA\rightarrow\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}(- 5\cdot rsin(r^{2}) -\frac{ (37\cdot rcos(\frac{(2\cdot r^{2})}{3}))}{5})drd\theta\\&\text{The bounds of our radius we can readily take from the problem. To find the angle ranges:}\end{align*}$

$\begin{align*}&\theta_1=arctan(\frac{0.750}{0.661})=0.27\pi;\theta_2=arctan(\frac{-0.063}{-0.998})=1.02\pi\\&\text{Now, utilizing integral rules:}\\&\int[rsin(ar^2)]=-\frac{cos(ar^2)}{2a}=-\frac{cos^2(\frac{a}{2}r^2)}{a}\\&\int[rcos(ar^2)]=\frac{sin(ar^2)}{2a}\\&\int_{0.27\pi}^{1.02\pi}\int_{0.36}^{1.33}(- 5\cdot rsin(r^{2}) -\frac{ (37\cdot rcos(\frac{(2\cdot r^{2})}{3}))}{5})drd\theta=(\frac{(5cos(r^{2}))}{2}-\frac{ (111sin(\frac{(2\cdot r^{2})}{3}))}{20})d\theta|_{0.36}^{1.33}\\&\int_{0.27\pi}^{1.02\pi}(-7.622)d\theta=(-7.622\theta)|_{0.27\pi}^{1.02\pi}=-17.96\end{align*}$

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Example Question #121 : Double Integration In Polar Coordinates

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(23e^{(- 2x^{2} - 2y^{2})} +\frac{ (3sin(\frac{(2x^{2})}{3}+\frac{ (2y^{2})}{3}))}{17})dA\text{, where D is}\\&\text{the region of a circle centered on the origin with a radius of }\\&1.91\\&\text{and encompassed by vectors from the origin }\\&\overrightarrow{u_1}=(0.309,0.951)\text{ and }\overrightarrow{u_2}=(-0.992,-0.125)\\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

Possible Answers:

12.02

-60.11

-6.01

Correct answer:

12.02

Explanation:

$\begin{align*}&\text{The current form of the integral is rather unwieldy, due to the }x^2\text{ and }y^2\text{ terms.}\\&\text{An approach that would be beneficial is a conversion to polar form:}\\&r=cos(\theta);r=sin(\theta)\\&r^2=x^2+y^2\\&dA=rdrd\theta\\&\text{With this we can find: }\iint_{D}(23e^{(- 2x^{2} - 2y^{2})} +\frac{ (3sin(\frac{(2x^{2})}{3}+\frac{ (2y^{2})}{3}))}{17})dA\rightarrow\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}(23\cdot re^{(-2\cdot r^{2})} +\frac{ (3\cdot rsin(\frac{(2\cdot r^{2})}{3}))}{17})drd\theta\\&\text{The bounds of our radius we can readily take from the problem. To find the angle ranges:}\end{align*}$

$\begin{align*}&\theta_1=arctan(\frac{0.951}{0.309})=0.4\pi;\theta_2=arctan(\frac{-0.125}{-0.992})=1.04\pi\\&\text{Now, utilizing integral rules:}\\&\int[rsin(ar^2)]=-\frac{cos(ar^2)}{2a}=-\frac{cos^2(\frac{a}{2}r^2)}{a}\\&\int[re^{ar^2}]=\frac{e^{ar^2}}{2a}\\&\int_{0.4\pi}^{1.04\pi}\int_{0}^{1.91}(23\cdot re^{(-2\cdot r^{2})} +\frac{ (3\cdot rsin(\frac{(2\cdot r^{2})}{3}))}{17})drd\theta=(-\frac{ (9cos(\frac{(2\cdot r^{2})}{3}))}{68}-\frac{ (23e^{(-2\cdot r^{2})})}{4})d\theta|_{0}^{1.91}\\&\int_{0.4\pi}^{1.04\pi}(5.979)d\theta=(5.979\theta)|_{0.4\pi}^{1.04\pi}=12.02\end{align*}$

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Example Question #602 : Multiple Integration

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(- 4e^{(-\frac{ (2x^{2})}{3}-\frac{ (2y^{2})}{3})} - 3\cdot 2^{(x^{2} + y^{2})})dA\text{, where D is}\\&\text{the region of a circle centered on the origin with a radius of }\\&1.2\\&\text{and encompassed by vectors from the origin }\\&\overrightarrow{u_1}=(-1.000,0.000)\text{ and }\overrightarrow{u_2}=(0.426,-0.905)\\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

Possible Answers:

-11.18

11.18

-44.71

5.59

Correct answer:

-11.18

Explanation:

$\begin{align*}&\text{The current form of the integral is rather unwieldy, due to the }x^2\text{ and }y^2\text{ terms.}\\&\text{An approach that would be beneficial is a conversion to polar form:}\\&r=cos(\theta);r=sin(\theta)\\&r^2=x^2+y^2\\&dA=rdrd\theta\\&\text{With this we can find: }\iint_{D}(- 4e^{(-\frac{ (2x^{2})}{3}-\frac{ (2y^{2})}{3})} - 3\cdot 2^{(x^{2} + y^{2})})dA\rightarrow\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}(- 3\cdot 2^{(r^{2})}\cdot r - 4\cdot re^{(-\frac{(2\cdot r^{2})}{3})})drd\theta\\&\text{The bounds of our radius we can readily take from the problem. To find the angle ranges:}\end{align*}$

$\begin{align*}&\theta_1=arctan(\frac{0.000}{-1.000})=1\pi;\theta_2=arctan(\frac{-0.905}{0.426})=1.64\pi\\&\text{Now, utilizing integral rules:}\\&\int[rb^{ar^2}]=\frac{b^{ar^2}}{2aln(b)}\\&\int[re^{ar^2}]=\frac{e^{ar^2}}{2a}\\&\int_{1\pi}^{1.64\pi}\int_{0}^{1.2}(- 3\cdot 2^{(r^{2})}\cdot r - 4\cdot re^{(-\frac{(2\cdot r^{2})}{3})})drd\theta=(3e^{(-\frac{(2\cdot r^{2})}{3})} -\frac{ (3\cdot 2^{(r^{2})})}{(2ln(2))})d\theta|_{0}^{1.2}\\&\int_{1\pi}^{1.64\pi}(-5.559)d\theta=(-5.559\theta)|_{1\pi}^{1.64\pi}=-11.18\end{align*}$

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Example Question #603 : Multiple Integration

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(\frac{(3cos(x^{2} + y^{2}))}{2}-\frac{ cos(2x^{2} + 2y^{2})}{8})dA\text{, where D is}\\&\text{the region defined between two circles centered on the origin with radii }\\&0.25\text{ and }1.93\\&\text{and encompassed by vectors from the origin }\\&\overrightarrow{u_1}=(0.218,0.976)\text{ and }\overrightarrow{u_2}=(-0.941,-0.339)\\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

Possible Answers:

0.17

-1.04

4.14

-0.52

Correct answer:

-1.04

Explanation:

$\begin{align*}&\text{The current form of the integral is rather unwieldy, due to the }x^2\text{ and }y^2\text{ terms.}\\&\text{An approach that would be beneficial is a conversion to polar form:}\\&r=cos(\theta);r=sin(\theta)\\&r^2=x^2+y^2\\&dA=rdrd\theta\\&\text{With this we can find: }\iint_{D}(\frac{(3cos(x^{2} + y^{2}))}{2}-\frac{ cos(2x^{2} + 2y^{2})}{8})dA\rightarrow\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}(\frac{(3\cdot rcos(r^{2}))}{2}-\frac{ (rcos(2\cdot r^{2}))}{8})drd\theta\\&\text{The bounds of our radius we can readily take from the problem. To find the angle ranges:}\end{align*}$

$\begin{align*}&\theta_1=arctan(\frac{0.976}{0.218})=0.43\pi;\theta_2=arctan(\frac{-0.339}{-0.941})=1.11\pi\\&\text{Now, utilizing integral rules:}\\&\int[rcos(ar^2)]=\frac{sin(ar^2)}{2a}\\&\int_{0.43\pi}^{1.11\pi}\int_{0.25}^{1.93}(\frac{(3\cdot rcos(r^{2}))}{2}-\frac{ (rcos(2\cdot r^{2}))}{8})drd\theta=(-\frac{(sin(r^{2})\cdot (cos(r^{2}) - 12))}{16})d\theta|_{0.25}^{1.93}\\&\int_{0.43\pi}^{1.11\pi}(-0.4848)d\theta=(-0.4848\theta)|_{0.43\pi}^{1.11\pi}=-1.04\end{align*}$

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Example Question #604 : Multiple Integration

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(\frac{(4e^{(- x^{2} - y^{2})})}{3}-\frac{ e^{(-\frac{ (3x^{2})}{2}-\frac{ (3y^{2})}{2})}}{2})dA\text{, where D is}\\&\text{the region defined between two circles centered on the origin with radii }\\&0.38\text{ and }1.96\\&\text{and encompassed by vectors from the origin }\\&\overrightarrow{u_1}=(-0.588,0.809)\text{ and }\overrightarrow{u_2}=(-0.397,-0.918)\\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

Possible Answers:

-0.45

0.9

-1.81

0.15

Correct answer:

0.9

Explanation:

$\begin{align*}&\text{The current form of the integral is rather unwieldy, due to the }x^2\text{ and }y^2\text{ terms.}\\&\text{An approach that would be beneficial is a conversion to polar form:}\\&r=cos(\theta);r=sin(\theta)\\&r^2=x^2+y^2\\&dA=rdrd\theta\\&\text{With this we can find: }\iint_{D}(\frac{(4e^{(- x^{2} - y^{2})})}{3}-\frac{ e^{(-\frac{ (3x^{2})}{2}-\frac{ (3y^{2})}{2})}}{2})dA\rightarrow\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}(\frac{(4\cdot re^{(-r^{2})})}{3}-\frac{ (re^{(-\frac{(3\cdot r^{2})}{2})})}{2})drd\theta\\&\text{The bounds of our radius we can readily take from the problem. To find the angle ranges:}\end{align*}$

$\begin{align*}&\theta_1=arctan(\frac{0.809}{-0.588})=0.7\pi;\theta_2=arctan(\frac{-0.918}{-0.397})=1.37\pi\\&\text{Now, utilizing integral rules:}\\&\int[re^{ar^2}]=\frac{e^{ar^2}}{2a}\\&\int_{0.7\pi}^{1.37\pi}\int_{0.38}^{1.96}(\frac{(4\cdot re^{(-r^{2})})}{3}-\frac{ (re^{(-\frac{(3\cdot r^{2})}{2})})}{2})drd\theta=(\frac{e^{(-\frac{(3\cdot r^{2})}{2})}}{6}-\frac{ (2e^{(-r^{2})})}{3})d\theta|_{0.38}^{1.96}\\&\int_{0.7\pi}^{1.37\pi}(0.429)d\theta=(0.429\theta)|_{0.7\pi}^{1.37\pi}=0.9\end{align*}$

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Example Question #121 : Double Integrals

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(7e^{(x^{2} + y^{2})} -\frac{ (\frac{46}{2^{(\frac{x^{2}}{2}+\frac{ y^{2}}{2})}})}{5})dA\text{, where D is}\\&\text{the region defined between two circles centered on the origin with radii }\\&0.43\text{ and }1.54\\&\text{and encompassed by vectors from the origin }\\&\overrightarrow{u_1}=(0.750,0.661)\text{ and }\overrightarrow{u_2}=(-0.905,-0.426)\\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

Possible Answers:

38.13

76.26

-19.07

-381.31

Correct answer:

76.26

Explanation:

$\begin{align*}&\text{The current form of the integral is rather unwieldy, due to the }x^2\text{ and }y^2\text{ terms.}\\&\text{An approach that would be beneficial is a conversion to polar form:}\\&r=cos(\theta);r=sin(\theta)\\&r^2=x^2+y^2\\&dA=rdrd\theta\\&\text{With this we can find: }\iint_{D}(7e^{(x^{2} + y^{2})} -\frac{ (\frac{46}{2^{(\frac{x^{2}}{2}+\frac{ y^{2}}{2})}})}{5})dA\rightarrow\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}(7\cdot re^{(r^{2})} -\frac{ (46\cdot r)}{(5\cdot 2^{(\frac{r^{2}}{2})})})drd\theta\\&\text{The bounds of our radius we can readily take from the problem. To find the angle ranges:}\end{align*}$

$\begin{align*}&\theta_1=arctan(\frac{0.661}{0.750})=0.23\pi;\theta_2=arctan(\frac{-0.426}{-0.905})=1.14\pi\\&\text{Now, utilizing integral rules:}\\&\int[re^{ar^2}]=\frac{e^{ar^2}}{2a}\\&\int[rb^{ar^2}]=\frac{b^{ar^2}}{2aln(b)}\\&\int_{0.23\pi}^{1.14\pi}\int_{0.43}^{1.54}(7\cdot re^{(r^{2})} -\frac{ (46\cdot r)}{(5\cdot 2^{(\frac{r^{2}}{2})})})drd\theta=(\frac{(7e^{(r^{2})})}{2}+\frac{ 46}{(5\cdot 2^{(\frac{r^{2}}{2})}ln(2))})d\theta|_{0.43}^{1.54}\\&\int_{0.23\pi}^{1.14\pi}(26.68)d\theta=(26.68\theta)|_{0.23\pi}^{1.14\pi}=76.26\end{align*}$

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Example Question #605 : Multiple Integration

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(- sin(\frac{x^{2}}{2}+\frac{ y^{2}}{2}) -\frac{ 17}{(6\cdot (x^{2} + y^{2}))})dA\text{, where D is}\\&\text{the region defined between two circles centered on the origin with radii }\\&0.46\text{ and }1.52\\&\text{and encompassed by vectors from the origin }\\&\overrightarrow{u_1}=(-0.976,0.218)\text{ and }\overrightarrow{u_2}=(0.613,-0.790)\\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

Possible Answers:

-9.75

48.73

-38.98

4.87

Correct answer:

-9.75

Explanation:

$\begin{align*}&\text{The current form of the integral is rather unwieldy, due to the }x^2\text{ and }y^2\text{ terms.}\\&\text{An approach that would be beneficial is a conversion to polar form:}\\&r=cos(\theta);r=sin(\theta)\\&r^2=x^2+y^2\\&dA=rdrd\theta\\&\text{With this we can find: }\iint_{D}(- sin(\frac{x^{2}}{2}+\frac{ y^{2}}{2}) -\frac{ 17}{(6\cdot (x^{2} + y^{2}))})dA\rightarrow\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}(- rsin(\frac{r^{2}}{2}) -\frac{ 17}{(6\cdot r)})drd\theta\\&\text{The bounds of our radius we can readily take from the problem. To find the angle ranges:}\end{align*}$

$\begin{align*}&\theta_1=arctan(\frac{0.218}{-0.976})=0.93\pi;\theta_2=arctan(\frac{-0.790}{0.613})=1.71\pi\\&\text{Now, utilizing integral rules:}\\&\int[rsin(ar^2)]=-\frac{cos(ar^2)}{2a}=-\frac{cos^2(\frac{a}{2}r^2)}{a}\\&\int[\frac{a}{r}]=aln(r)=ln(r^a)\\&\int_{0.93\pi}^{1.71\pi}\int_{0.46}^{1.52}(- rsin(\frac{r^{2}}{2}) -\frac{ 17}{(6\cdot r)})drd\theta=(cos(\frac{r^{2}}{2}) -\frac{ (17ln(r))}{6})d\theta|_{0.46}^{1.52}\\&\int_{0.93\pi}^{1.71\pi}(-3.977)d\theta=(-3.977\theta)|_{0.93\pi}^{1.71\pi}=-9.75\end{align*}$

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Example Question #121 : Double Integrals

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(\frac{e^{(- 2x^{2} - 2y^{2})}}{3}+ (\frac{\frac{2}{3})^{(\frac{x^{2}}{2}+\frac{ y^{2}}{2})}}{4})dA\text{, where D is}\\&\text{the region of a circle centered on the origin with a radius of }\\&1.37\\&\text{and encompassed by vectors from the origin }\\&\overrightarrow{u_1}=(0.482,0.876)\text{ and }\overrightarrow{u_2}=(0.861,-0.509)\\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

Possible Answers:

-0.26

0.26

-5.18

1.29

Correct answer:

1.29

Explanation:

$\begin{align*}&\text{The current form of the integral is rather unwieldy, due to the }x^2\text{ and }y^2\text{ terms.}\\&\text{An approach that would be beneficial is a conversion to polar form:}\\&r=cos(\theta);r=sin(\theta)\\&r^2=x^2+y^2\\&dA=rdrd\theta\\&\text{With this we can find: }\iint_{D}(\frac{e^{(- 2x^{2} - 2y^{2})}}{3}+ (\frac{\frac{2}{3})^{(\frac{x^{2}}{2}+\frac{ y^{2}}{2})}}{4})dA\rightarrow\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}(\frac{((\frac{2}{3})^{(\frac{r^{2}}{2})}\cdot r)}{4}+\frac{ (re^{(-2\cdot r^{2})})}{3})drd\theta\\&\text{The bounds of our radius we can readily take from the problem. To find the angle ranges:}\end{align*}$

$\begin{align*}&\theta_1=arctan(\frac{0.876}{0.482})=0.34\pi;\theta_2=arctan(\frac{-0.509}{0.861})=1.83\pi\\&\text{Now, utilizing integral rules:}\\&\int[re^{ar^2}]=\frac{e^{ar^2}}{2a}\\&\int[rb^{ar^2}]=\frac{b^{ar^2}}{2aln(b)}\\&\int_{0.34\pi}^{1.83\pi}\int_{0}^{1.37}(\frac{((\frac{2}{3})^{(\frac{r^{2}}{2})}\cdot r)}{4}+\frac{ (re^{(-2\cdot r^{2})})}{3})drd\theta=((\frac{\frac{2}{3})^{(\frac{r^{2}}{2})}}{(4ln(\frac{2}{3}))}-\frac{ e^{(-2\cdot r^{2})}}{12})d\theta|_{0}^{1.37}\\&\int_{0.34\pi}^{1.83\pi}(0.2765)d\theta=(0.2765\theta)|_{0.34\pi}^{1.83\pi}=1.29\end{align*}$

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Example Question #607 : Multiple Integration

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(\frac{(5e^{(- x^{2} - y^{2})})}{14}- 49cos(2x^{2} + 2y^{2}))dA\text{, where D is}\\&\text{the region of a circle centered on the origin with a radius of }\\&0.72\\&\text{and encompassed by vectors from the origin }\\&\overrightarrow{u_1}=(0.536,0.844)\text{ and }\overrightarrow{u_2}=(0.309,-0.951)\\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

Possible Answers:

84.22

-10.53

-42.11

14.04

Correct answer:

-42.11

Explanation:

$\begin{align*}&\text{The current form of the integral is rather unwieldy, due to the }x^2\text{ and }y^2\text{ terms.}\\&\text{An approach that would be beneficial is a conversion to polar form:}\\&r=cos(\theta);r=sin(\theta)\\&r^2=x^2+y^2\\&dA=rdrd\theta\\&\text{With this we can find: }\iint_{D}(\frac{(5e^{(- x^{2} - y^{2})})}{14}- 49cos(2x^{2} + 2y^{2}))dA\rightarrow\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}(\frac{(5\cdot re^{(-r^{2})})}{14}- 49\cdot rcos(2\cdot r^{2}))drd\theta\\&\text{The bounds of our radius we can readily take from the problem. To find the angle ranges:}\end{align*}$

$\begin{align*}&\theta_1=arctan(\frac{0.844}{0.536})=0.32\pi;\theta_2=arctan(\frac{-0.951}{0.309})=1.6\pi\\&\text{Now, utilizing integral rules:}\\&\int[rcos(ar^2)]=\frac{sin(ar^2)}{2a}\\&\int[re^{ar^2}]=\frac{e^{ar^2}}{2a}\\&\int_{0.32\pi}^{1.6\pi}\int_{0}^{0.72}(\frac{(5\cdot re^{(-r^{2})})}{14}- 49\cdot rcos(2\cdot r^{2}))drd\theta=(-\frac{ (5e^{(-r^{2})})}{28}-\frac{ (49sin(2\cdot r^{2}))}{4})d\theta|_{0}^{0.72}\\&\int_{0.32\pi}^{1.6\pi}(-10.47)d\theta=(-10.47\theta)|_{0.32\pi}^{1.6\pi}=-42.11\end{align*}$

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Calculus 3 : Multiple Integration

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