- #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?