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Calculus 3 : Double Integrals

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Example Questions

Example Question #21 : Double Integration In Polar Coordinates

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(24sin(\frac{x^{2}}{2}+\frac{ y^{2}}{2}))dxdy\text{, where D is}\\&\text{the region defined between two circles centered on the origin with radii }\\&0.26\text{ and }1.93\\&\text{and encompassed by vectors from the origin }\\&\overrightarrow{u_1}=(-0.729,0.685)\text{ and }\overrightarrow{u_2}=(0.976,-0.218)\\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

Possible Answers:

-567.65

37.84

113.53

28.38

Correct answer:

113.53

Explanation:

$\begin{align*}&\text{The current form of the integral is rather unwieldy, due to the }x^2\text{ and }y^2\text{ terms.}\\&\text{An approach that would be beneficial is a conversion to polar form:}\\&r=cos(\theta);r=sin(\theta)\\&r^2=x^2+y^2\\&dxdy=rdrd\theta\\&\text{With this we can find: }\iint_{D}(24sin(\frac{x^{2}}{2}+\frac{ y^{2}}{2}))dxdy\rightarrow\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}(24\cdot rsin(\frac{r^{2}}{2}))drd\theta\\&\text{The bounds of our radius we can readily take from the problem. To find the angle ranges:}\end{align*}$

$\begin{align*}&\theta_1=arctan(\frac{0.685}{-0.729})=0.76\pi;\theta_2=arctan(\frac{-0.218}{0.976})=1.93\pi\\&\text{Now, utilizing integral rules:}\\&\int[rsin(ar^2)]=-\frac{cos(ar^2)}{2a}=-\frac{cos^2(\frac{a}{2}r^2)}{a}\\&\int_{0.76\pi}^{1.93\pi}\int_{0.26}^{1.93}(24\cdot rsin(\frac{r^{2}}{2}))drd\theta=(-24cos(\frac{r^{2}}{2}))d\theta|_{0.26}^{1.93}\\&\int_{0.76\pi}^{1.93\pi}(30.89)d\theta=(30.89\cdot \theta)|_{0.76\pi}^{1.93\pi}=113.53\end{align*}$

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Example Question #22 : Double Integration In Polar Coordinates

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(-7cos(x^{2} + y^{2}))dxdy\text{, where D is}\\&\text{the region defined between two circles centered on the origin with radii }\\&0.19\text{ and }1.48\\&\text{and encompassed by vectors from the origin }\\&\overrightarrow{u_1}=(0.000,1.000)\text{ and }\overrightarrow{u_2}=(0.454,-0.891)\\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

Possible Answers:

9.84

-2.46

-9.84

-59.03

Correct answer:

-9.84

Explanation:

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

$\begin{align*}&\theta_1=arctan(\frac{1.000}{0.000})=0.5\pi;\theta_2=arctan(\frac{-0.891}{0.454})=1.65\pi\\&\text{Now, utilizing integral rules:}\\&\int[rcos(ar^2)]=\frac{sin(ar^2)}{2a}\\&\int_{0.5\pi}^{1.65\pi}\int_{0.19}^{1.48}(-7\cdot rcos(r^{2}))drd\theta=(-\frac{(7sin(r^{2}))}{2})d\theta|_{0.19}^{1.48}\\&\int_{0.5\pi}^{1.65\pi}(-2.723)d\theta=(-2.723\cdot \theta)|_{0.5\pi}^{1.65\pi}=-9.84\end{align*}$

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Example Question #23 : Double Integration In Polar Coordinates

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(\frac{(21\cdot (\frac{3}{2})^{(x^{2} + y^{2})})}{2})dxdy\text{, where D is}\\&\text{the region of a circle centered on the origin with a radius of }\\&0.97\\&\text{and encompassed by vectors from the origin }\\&\overrightarrow{u_1}=(0.588,0.809)\text{ and }\overrightarrow{u_2}=(-0.661,-0.750)\\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

Possible Answers:

4.58

18.33

-18.33

6.11

Correct answer:

18.33

Explanation:

$\begin{align*}&\text{The current form of the integral is rather unwieldy, due to the }x^2\text{ and }y^2\text{ terms.}\\&\text{An approach that would be beneficial is a conversion to polar form:}\\&r=cos(\theta);r=sin(\theta)\\&r^2=x^2+y^2\\&dxdy=rdrd\theta\\&\text{With this we can find: }\iint_{D}(\frac{(21\cdot (\frac{3}{2})^{(x^{2} + y^{2})})}{2})dxdy\rightarrow\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}(\frac{(21\cdot (\frac{3}{2})^{(r^{2})}\cdot r)}{2})drd\theta\\&\text{The bounds of our radius we can readily take from the problem. To find the angle ranges:}\end{align*}$

$\begin{align*}&\theta_1=arctan(\frac{0.809}{0.588})=0.3\pi;\theta_2=arctan(\frac{-0.750}{-0.661})=1.27\pi\\&\text{Now, utilizing integral rules:}\\&\int[rb^{ar^2}]=\frac{b^{ar^2}}{2aln(b)}\\&\int_{0.3\pi}^{1.27\pi}\int_{0}^{0.97}(\frac{(21\cdot (\frac{3}{2})^{(r^{2})}\cdot r)}{2})drd\theta=(\frac{(21\cdot (\frac{3}{2})^{(r^{2})})}{(4ln(\frac{3}{2}))})d\theta|_{0}^{0.97}\\&\int_{0.3\pi}^{1.27\pi}(6.014)d\theta=(6.014\cdot \theta)|_{0.3\pi}^{1.27\pi}=18.33\end{align*}$

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Example Question #24 : Double Integration In Polar Coordinates

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(\frac{(4e^{(-\frac{ x^{2}}{2}-\frac{ y^{2}}{2})})}{37})dxdy\text{, where D is}\\&\text{the region defined between two circles centered on the origin with radii }\\&0.24\text{ and }0.84\\&\text{and encompassed by vectors from the origin }\\&\overrightarrow{u_1}=(0.996,0.094)\text{ and }\overrightarrow{u_2}=(-0.992,0.125)\\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

Possible Answers:

-0.51

0.08

0.42

0.01

Correct answer:

0.08

Explanation:

$\begin{align*}&\text{The current form of the integral is rather unwieldy, due to the }x^2\text{ and }y^2\text{ terms.}\\&\text{An approach that would be beneficial is a conversion to polar form:}\\&r=cos(\theta);r=sin(\theta)\\&r^2=x^2+y^2\\&dxdy=rdrd\theta\\&\text{With this we can find: }\iint_{D}(\frac{(4e^{(-\frac{ x^{2}}{2}-\frac{ y^{2}}{2})})}{37})dxdy\rightarrow\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}(\frac{(4\cdot re^{(-\frac{r^{2}}{2})})}{37})drd\theta\\&\text{The bounds of our radius we can readily take from the problem. To find the angle ranges:}\end{align*}$

$\begin{align*}&\theta_1=arctan(\frac{0.094}{0.996})=0.03\pi;\theta_2=arctan(\frac{0.125}{-0.992})=0.96\pi\\&\text{Now, utilizing integral rules:}\\&\int[re^{ar^2}]=\frac{e^{ar^2}}{2a}\\&\int_{0.03\pi}^{0.96\pi}\int_{0.24}^{0.84}(\frac{(4\cdot re^{(-\frac{r^{2}}{2})})}{37})drd\theta=(-\frac{(4e^{(-\frac{r^{2}}{2})})}{37})d\theta|_{0.24}^{0.84}\\&\int_{0.03\pi}^{0.96\pi}(0.02907)d\theta=(0.02907\cdot \theta)|_{0.03\pi}^{0.96\pi}=0.08\end{align*}$

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Example Question #25 : Double Integrals

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(\frac{(2sin(\frac{(3x^{2})}{2}+\frac{ (3y^{2})}{2}))}{37})dxdy\text{, where D is}\\&\text{the region defined between two circles centered on the origin with radii }\\&0.41\text{ and }1.85\\&\text{and encompassed by vectors from the origin }\\&\overrightarrow{u_1}=(-0.063,0.998)\text{ and }\overrightarrow{u_2}=(0.156,-0.988)\\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

Possible Answers:

-0.16

0.2

0.08

0.03

Correct answer:

0.03

Explanation:

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

$\begin{align*}&\theta_1=arctan(\frac{0.998}{-0.063})=0.52\pi;\theta_2=arctan(\frac{-0.988}{0.156})=1.55\pi\\&\text{Now, utilizing integral rules:}\\&\int[rsin(ar^2)]=-\frac{cos(ar^2)}{2a}=-\frac{cos^2(\frac{a}{2}r^2)}{a}\\&\int_{0.52\pi}^{1.55\pi}\int_{0.41}^{1.85}(\frac{(2\cdot rsin(\frac{(3\cdot r^{2})}{2}))}{37})drd\theta=(-\frac{(2cos(\frac{(3\cdot r^{2})}{2}))}{111})d\theta|_{0.41}^{1.85}\\&\int_{0.52\pi}^{1.55\pi}(0.01008)d\theta=(0.01008\cdot \theta)|_{0.52\pi}^{1.55\pi}=0.03\end{align*}$

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Example Question #26 : Double Integrals

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(-4sin(\frac{x^{2}}{2}+\frac{ y^{2}}{2}))dxdy\text{, where D is}\\&\text{the region of a circle centered on the origin with a radius of }\\&1.97\\&\text{and encompassed by vectors from the origin }\\&\overrightarrow{u_1}=(-0.685,0.729)\text{ and }\overrightarrow{u_2}=(0.218,-0.976)\\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

Possible Answers:

-3.55

-14.2

85.19

-70.99

Correct answer:

-14.2

Explanation:

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

$\begin{align*}&\theta_1=arctan(\frac{0.729}{-0.685})=0.74\pi;\theta_2=arctan(\frac{-0.976}{0.218})=1.57\pi\\&\text{Now, utilizing integral rules:}\\&\int[rsin(ar^2)]=-\frac{cos(ar^2)}{2a}=-\frac{cos^2(\frac{a}{2}r^2)}{a}\\&\int_{0.74\pi}^{1.57\pi}\int_{0}^{1.97}(-4\cdot rsin(\frac{r^{2}}{2}))drd\theta=(4cos(\frac{r^{2}}{2}))d\theta|_{0}^{1.97}\\&\int_{0.74\pi}^{1.57\pi}(-5.445)d\theta=(-5.445\cdot \theta)|_{0.74\pi}^{1.57\pi}=-14.2\end{align*}$

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Example Question #27 : Double Integrals

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(\frac{(47sin(x^{2} + y^{2}))}{2}-\frac{ e^{(x^{2} + y^{2})}}{4})dxdy\text{, where D is}\\&\text{the region of a circle centered on the origin with a radius of }\\&1.14\\&\text{and encompassed by vectors from the origin }\\&\overrightarrow{u_1}=(0.969,0.249)\text{ and }\overrightarrow{u_2}=(-0.988,0.156)\\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

Possible Answers:

67.8

22.6

-90.4

-11.3

Correct answer:

22.6

Explanation:

$\begin{align*}&\text{The current form of the integral is rather unwieldy, due to the }x^2\text{ and }y^2\text{ terms.}\\&\text{An approach that would be beneficial is a conversion to polar form:}\\&r=cos(\theta);r=sin(\theta)\\&r^2=x^2+y^2\\&dxdy=rdrd\theta\\&\text{With this we can find: }\iint_{D}(\frac{(47sin(x^{2} + y^{2}))}{2}-\frac{ e^{(x^{2} + y^{2})}}{4})dxdy\rightarrow\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}(\frac{(47\cdot rsin(r^{2}))}{2}-\frac{ (re^{(r^{2})})}{4})drd\theta\\&\text{The bounds of our radius we can readily take from the problem. To find the angle ranges:}\end{align*}$

$\begin{align*}&\theta_1=arctan(\frac{0.249}{0.969})=0.08\pi;\theta_2=arctan(\frac{0.156}{-0.988})=0.95\pi\\&\text{Now, utilizing integral rules:}\\&\int[rsin(ar^2)]=-\frac{cos(ar^2)}{2a}=-\frac{cos^2(\frac{a}{2}r^2)}{a}\\&\int[re^{ar^2}]=\frac{e^{ar^2}}{2a}\\&\int_{0.08\pi}^{0.95\pi}\int_{0}^{1.14}(\frac{(47\cdot rsin(r^{2}))}{2}-\frac{ (re^{(r^{2})})}{4})drd\theta=(-\frac{ (47cos(r^{2}))}{4}-\frac{ e^{(r^{2})}}{8})d\theta|_{0}^{1.14}\\&\int_{0.08\pi}^{0.95\pi}(8.269)d\theta=(8.269\cdot \theta)|_{0.08\pi}^{0.95\pi}=22.6\end{align*}$

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Example Question #28 : Double Integrals

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(\frac{(5e^{(-\frac{ x^{2}}{2}-\frac{ y^{2}}{2})})}{44}- 8e^{(\frac{x^{2}}{2}+\frac{ y^{2}}{2})})dxdy\text{, where D is}\\&\text{the region of a circle centered on the origin with a radius of }\\&1.62\\&\text{and encompassed by vectors from the origin }\\&\overrightarrow{u_1}=(-0.930,0.368)\text{ and }\overrightarrow{u_2}=(0.509,-0.861)\\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

Possible Answers:

-53.69

-10.74

161.06

8.95

Correct answer:

-53.69

Explanation:

$\begin{align*}&\text{The current form of the integral is rather unwieldy, due to the }x^2\text{ and }y^2\text{ terms.}\\&\text{An approach that would be beneficial is a conversion to polar form:}\\&r=cos(\theta);r=sin(\theta)\\&r^2=x^2+y^2\\&dxdy=rdrd\theta\\&\text{With this we can find: }\iint_{D}(\frac{(5e^{(-\frac{ x^{2}}{2}-\frac{ y^{2}}{2})})}{44}- 8e^{(\frac{x^{2}}{2}+\frac{ y^{2}}{2})})dxdy\rightarrow\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}(\frac{(5\cdot re^{(-\frac{r^{2}}{2})})}{44}- 8\cdot re^{(\frac{r^{2}}{2})})drd\theta\\&\text{The bounds of our radius we can readily take from the problem. To find the angle ranges:}\end{align*}$

$\begin{align*}&\theta_1=arctan(\frac{0.368}{-0.930})=0.88\pi;\theta_2=arctan(\frac{-0.861}{0.509})=1.67\pi\\&\text{Now, utilizing integral rules:}\\&\int[re^{ar^2}]=\frac{e^{ar^2}}{2a}\\&\int_{0.88\pi}^{1.67\pi}\int_{0}^{1.62}(\frac{(5\cdot re^{(-\frac{r^{2}}{2})})}{44}- 8\cdot re^{(\frac{r^{2}}{2})})drd\theta=(-\frac{(e^{(-\frac{r^{2}}{2})}\cdot (352e^{(r^{2})} + 5))}{44})d\theta|_{0}^{1.62}\\&\int_{0.88\pi}^{1.67\pi}(-21.63)d\theta=(-21.63\cdot \theta)|_{0.88\pi}^{1.67\pi}=-53.69\end{align*}$

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Example Question #1401 : Calculus 3

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(-\frac{ cos(\frac{(2x^{2})}{3}+\frac{ (2y^{2})}{3})}{24}-\frac{ sin(\frac{(2x^{2})}{3}+\frac{ (2y^{2})}{3})}{43})dxdy\text{, where D is}\\&\text{the region of a circle centered on the origin with a radius of }\\&1.63\\&\text{and encompassed by vectors from the origin }\\&\overrightarrow{u_1}=(-0.454,0.891)\text{ and }\overrightarrow{u_2}=(0.063,-0.998)\\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

Possible Answers:

0.07

-0.05

0.42

-0.14

Correct answer:

-0.14

Explanation:

$\begin{align*}&\text{The current form of the integral is rather unwieldy, due to the }x^2\text{ and }y^2\text{ terms.}\\&\text{An approach that would be beneficial is a conversion to polar form:}\\&r=cos(\theta);r=sin(\theta)\\&r^2=x^2+y^2\\&dxdy=rdrd\theta\\&\text{With this we can find: }\iint_{D}(-\frac{ cos(\frac{(2x^{2})}{3}+\frac{ (2y^{2})}{3})}{24}-\frac{ sin(\frac{(2x^{2})}{3}+\frac{ (2y^{2})}{3})}{43})dxdy\rightarrow\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}(-\frac{ (rcos(\frac{(2\cdot r^{2})}{3}))}{24}-\frac{ (rsin(\frac{(2\cdot r^{2})}{3}))}{43})drd\theta\\&\text{The bounds of our radius we can readily take from the problem. To find the angle ranges:}\end{align*}$

$\begin{align*}&\theta_1=arctan(\frac{0.891}{-0.454})=0.65\pi;\theta_2=arctan(\frac{-0.998}{0.063})=1.52\pi\\&\text{Now, utilizing integral rules:}\\&\int[rsin(ar^2)]=-\frac{cos(ar^2)}{2a}=-\frac{cos^2(\frac{a}{2}r^2)}{a}\\&\int[rcos(ar^2)]=\frac{sin(ar^2)}{2a}\\&\int_{0.65\pi}^{1.52\pi}\int_{0}^{1.63}(-\frac{ (rcos(\frac{(2\cdot r^{2})}{3}))}{24}-\frac{ (rsin(\frac{(2\cdot r^{2})}{3}))}{43})drd\theta=(\frac{(3cos(\frac{(2\cdot r^{2})}{3}))}{172}-\frac{ sin(\frac{(2\cdot r^{2})}{3})}{32})d\theta|_{0}^{1.63}\\&\int_{0.65\pi}^{1.52\pi}(-0.05154)d\theta=(-0.05154\cdot \theta)|_{0.65\pi}^{1.52\pi}=-0.14\end{align*}$

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Example Question #1402 : Calculus 3

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(\frac{(5e^{(\frac{(2x^{2})}{3}+\frac{ (2y^{2})}{3})})}{9}+\frac{ 1}{(41\cdot (\frac{(2x^{2})}{3}+\frac{ (2y^{2})}{3}))})dxdy\text{, where D is}\\&\text{the region defined between two circles centered on the origin with radii }\\&0.22\text{ and }1.05\\&\text{and encompassed by vectors from the origin }\\&\overrightarrow{u_1}=(-0.998,0.063)\text{ and }\overrightarrow{u_2}=(0.918,-0.397)\\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

Possible Answers:

1.39

6.93

-2.77

-0.46

Correct answer:

1.39

Explanation:

$\begin{align*}&\text{The current form of the integral is rather unwieldy, due to the }x^2\text{ and }y^2\text{ terms.}\\&\text{An approach that would be beneficial is a conversion to polar form:}\\&r=cos(\theta);r=sin(\theta)\\&r^2=x^2+y^2\\&dxdy=rdrd\theta\\&\text{With this we can find: }\iint_{D}(\frac{(5e^{(\frac{(2x^{2})}{3}+\frac{ (2y^{2})}{3})})}{9}+\frac{ 1}{(41\cdot (\frac{(2x^{2})}{3}+\frac{ (2y^{2})}{3}))})dxdy\rightarrow\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}(\frac{(5\cdot re^{(\frac{(2\cdot r^{2})}{3})})}{9}+\frac{ 3}{(82\cdot r)})drd\theta\\&\text{The bounds of our radius we can readily take from the problem. To find the angle ranges:}\end{align*}$

$\begin{align*}&\theta_1=arctan(\frac{0.063}{-0.998})=0.98\pi;\theta_2=arctan(\frac{-0.397}{0.918})=1.87\pi\\&\text{Now, utilizing integral rules:}\\&\int[\frac{a}{r}]=aln(r)=ln(r^a)\\&\int[re^{ar^2}]=\frac{e^{ar^2}}{2a}\\&\int_{0.98\pi}^{1.87\pi}\int_{0.22}^{1.05}(\frac{(5\cdot re^{(\frac{(2\cdot r^{2})}{3})})}{9}+\frac{ 3}{(82\cdot r)})drd\theta=(\frac{(5e^{(\frac{(2\cdot r^{2})}{3})})}{12}+\frac{ (3ln(r))}{82})d\theta|_{0.22}^{1.05}\\&\int_{0.98\pi}^{1.87\pi}(0.4958)d\theta=(0.4958\cdot \theta)|_{0.98\pi}^{1.87\pi}=1.39\end{align*}$

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Calculus 3 : Double Integrals

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #21 : Double Integration In Polar Coordinates

Example Question #22 : Double Integration In Polar Coordinates

Example Question #23 : Double Integration In Polar Coordinates

Example Question #24 : Double Integration In Polar Coordinates

Example Question #25 : Double Integrals

Example Question #26 : Double Integrals

Example Question #27 : Double Integrals

Example Question #28 : Double Integrals

Example Question #1401 : Calculus 3

Example Question #1402 : Calculus 3

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