- #1

DottZakapa

- 239

- 17

- Homework Statement
- Let ## E=\left\{ (x,y,z) \in R^3 : 1 \leq x^2+y^2+z^2 \leq 4, 3x^2+3y^2-z^2\leq 0, z\geq0 \right\} ##

- Represent the region E in 3-dimensions

-represent the section of e in (x,z) plane

-compute ## \int \frac {y^2} {x^2+y^2} \,dx \,dy \,dz##

- Relevant Equations
- integrals

Let ## E=\left\{ (x,y,z) \in R^3 : 1 \leq x^2+y^2+z^2 \leq 4, 3x^2+3y^2-z^2\leq 0, z\geq0 \right\} ##

- Represent the region E in 3-dimensions

-represent the section of e in (x,z) plane

-compute ## \int \frac {y^2} {x^2+y^2} \,dx \,dy \,dz##

the domain is a sphere of radius 2 with an inner spherical hole of radius 1 which intersects a cone on the positive z-axis.

using spherical coordinates

##\begin{cases}

x=r cos\theta sin \phi\\

y=rsin\theta sin\phi\\

z=rcos\phi\\

\end{cases}##

##\begin{cases}

1\leq r\leq 2\\

0 \leq \theta \leq 2\pi\\

0 \leq \phi \leq \frac {\pi} 6\\

\end{cases}##

the integral becomes

## \int_{1}^2 \int_{0}^{2\pi} \int_{0}^{\frac \pi 6} \frac {(rsin\theta sin\phi)^2} {(r cos\theta sin \phi)^2+(rsin\theta sin\phi)^2} r^2 \sin \phi \,d\phi \, d\theta \,dr ##=

= ## \int_{1}^2 \int_{0}^{2\pi} \int_{0}^{\frac \pi 6} r^2 sin\phi sin\theta^2 ,d\phi \, d\theta \,dr##

up to here is correct?

- Represent the region E in 3-dimensions

-represent the section of e in (x,z) plane

-compute ## \int \frac {y^2} {x^2+y^2} \,dx \,dy \,dz##

the domain is a sphere of radius 2 with an inner spherical hole of radius 1 which intersects a cone on the positive z-axis.

using spherical coordinates

##\begin{cases}

x=r cos\theta sin \phi\\

y=rsin\theta sin\phi\\

z=rcos\phi\\

\end{cases}##

##\begin{cases}

1\leq r\leq 2\\

0 \leq \theta \leq 2\pi\\

0 \leq \phi \leq \frac {\pi} 6\\

\end{cases}##

the integral becomes

## \int_{1}^2 \int_{0}^{2\pi} \int_{0}^{\frac \pi 6} \frac {(rsin\theta sin\phi)^2} {(r cos\theta sin \phi)^2+(rsin\theta sin\phi)^2} r^2 \sin \phi \,d\phi \, d\theta \,dr ##=

= ## \int_{1}^2 \int_{0}^{2\pi} \int_{0}^{\frac \pi 6} r^2 sin\phi sin\theta^2 ,d\phi \, d\theta \,dr##

up to here is correct?