Calculus 3 - Triple Integration in Cylindrical Coordinates

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(3e^{(- 2x^{2} - 2y^{2})} -\frac{ cos(\frac{x^{2}}{2}+\frac{ y^{2}}{2})}{(11\cdot 2^{(\frac{z}{2})})})dA\text{, where D is}\&\text{the region of a cylinder centered on the origin with a radius of }\&0.84\&\text{and length }1.77\&\text{and encompassed by vectors from the origin }\&\overrightarrow{u_1}=(0.809,0.588)\text{ and }\overrightarrow{u_2}=(-0.960,-0.279)\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(-10cos(z + 1)sin(2x^{2} + 2y^{2}))dA\text{, where D is}\&\text{the region of a cylinder centered on the origin with a radius of }\&1.22\&\text{and length }1.91\&\text{and encompassed by vectors from the origin }\&\overrightarrow{u_1}=(-0.536,0.844)\text{ and }\overrightarrow{u_2}=(0.637,-0.771)\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(\frac{42}{2^{(\frac{(2x^{2})}{3}+\frac{ (2y^{2})}{3})}\cdot 2^z})dA\text{, where D is}\&\text{the region of a cylindrical annulus centered on the origin,}\&\text{with inner/outer radii }\&0.43\text{ and }1.71\&\text{and length }0.72\&\text{and encompassed by vectors from the origin }\&\overrightarrow{u_1}=(-0.063,0.998)\text{ and }\overrightarrow{u_2}=(-0.941,-0.339)\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(\frac{(49e^{(2z)}cos(x^{2} + y^{2}))}{6})dA\text{, where D is}\&\text{the region of a cylinder centered on the origin with a radius of }\&1.87\&\text{and length }1.45\&\text{and encompassed by vectors from the origin }\&\overrightarrow{u_1}=(-0.844,0.536)\text{ and }\overrightarrow{u_2}=(0.982,-0.187)\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(-\frac{(6e^{(-\frac{ (2x^{2})}{3}-\frac{ (2y^{2})}{3})}e^{(-z)})}{29})dA\text{, where D is}\&\text{the region of a cylindrical annulus centered on the origin,}\&\text{with inner/outer radii }\&0.25\text{ and }1.23\&\text{and length }1.8\&\text{and encompassed by vectors from the origin }\&\overrightarrow{u_1}=(-0.891,0.454)\text{ and }\overrightarrow{u_2}=(0.930,-0.368)\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(11cos(\frac{(3x^{2})}{2}+\frac{ (3y^{2})}{2})sin(3z))dA\text{, where D is}\&\text{the region of a cylinder centered on the origin with a radius of }\&1.31\&\text{and length }0.65\&\text{and encompassed by vectors from the origin }\&\overrightarrow{u_1}=(1.000,0.031)\text{ and }\overrightarrow{u_2}=(-0.976,-0.218)\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(\frac{(44\cdot 3^{(\frac{z}{2})}e^{(\frac{(3x^{2})}{2}+\frac{ (3y^{2})}{2})})}{5})dA\text{, where D is}\&\text{the region of a cylinder centered on the origin with a radius of }\&1.82\&\text{and length }1.21\&\text{and encompassed by vectors from the origin }\&\overrightarrow{u_1}=(0.809,0.588)\text{ and }\overrightarrow{u_2}=(-0.988,0.156)\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(-\frac{(e^{(2z)}sin(x^{2} + y^{2}))}{3})dA\text{, where D is}\&\text{the region of a cylinder centered on the origin with a radius of }\&1.11\&\text{and length }1.19\&\text{and encompassed by vectors from the origin }\&\overrightarrow{u_1}=(-0.426,0.905)\text{ and }\overrightarrow{u_2}=(-0.613,-0.790)\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(e^{(- x^{2} - y^{2})}\cdot (\frac{(5z)}{8}+\frac{ 5}{8}))dA\text{, where D is}\&\text{the region of a cylinder centered on the origin with a radius of }\&1.56\&\text{and length }1.4\&\text{and encompassed by vectors from the origin }\&\overrightarrow{u_1}=(-0.187,0.982)\text{ and }\overrightarrow{u_2}=(-0.729,-0.685)\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

$\begin{align*}&\text{Evaluate the integral, }\iint_{D}(-\frac{ (3cos(\frac{(3x^{2})}{2}+\frac{ (3y^{2})}{2}))}{41}-\frac{ (sin(z + 1)cos(x^{2} + y^{2}))}{25})dA\text{, where D is}\&\text{the region of a cylindrical annulus centered on the origin with radii }\&0.23\text{ and }1.56\&\text{and length }1.97\&\text{and encompassed by vectors from the origin }\&\overrightarrow{u_1}=(-0.094,0.996)\text{ and }\overrightarrow{u_2}=(0.637,-0.771)\&\text{counterclockwise from }\overrightarrow{u_1}\end{align*}$

Triple Integration in Cylindrical Coordinates

Practice Questions