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Sagot :
Evaluation of E(x+y+z)dV where E is the solid in the first octant that lies under the paraboloid z = 4 - x² - y² is [tex]\frac{128}{15} + \frac{8\pi}{3}[/tex]
Cylindrical coordinates are a set of three coordinates used to locate a point in the cylindrical coordinate system.
When the polar coordinates are extended to a three-dimensional plane, an additional z coordinate is added.
These three measurements combine to form cylindrical coordinates. The coordinates describe two distances and one angle.
According to the question,
We have to find the area under paraboloid
Given : E is a region that lies under paraboloid z = 4 - x² - y² and in the first octant.
The intersection of this paraboloid with the xy plane will give z = 0,
z = 4 - x² - y² , z = 0, that is a circle x² +y² = 4 with radius 2
Therefore , 0 ≤ z ≤ 4 - r²
Therefore, the region E in cylindrical coordinates is
[tex]E = ( r , \theta , z ) | 0 \leq \theta \geq \frac{\pi}{2} , 0 \leq 4 \geq , 0 \leq z \geq 4 - r^2[/tex]
f(x , y ,z ) = x + y + z
=> f(x , y ,z ) = rcosθ + rsinθ + z
=> f(x , y ,z ) = r(cosθ + sinθ) + z
Integration ,
[tex]{\int \int \int }\limits_E {z} \,dv[/tex]
=> [tex]\int\limits^{\frac{\pi}{2}}_0 \int\limits^{2}_0 \int\limits^{4 - r^2}_0 {[r ( cos \theta + sin \theta) + z] r \, dzdrd\theta[/tex]
simlifying,
[tex]= \int\limits^{\frac{\pi}{2}}_0 \int\limits^{2}_0 \int\limits^{4 - r^2}_0 {[r^2 ( cos \theta + sin \theta) + rz] \, dzdrd\theta\\= \int\limits^{\frac{\pi}{2}}_0 \int\limits^{2}_0 [r^2 ( cos \theta + sin \theta)z + r\frac{z^2}{2}] \, drd\theta[/tex]
=> [tex]\int\limits^{\frac{\pi}{2}}_0 \int\limits^{2}_0 [(4r^{2} - r^{4})( cos \theta + sin \theta) + \frac{r(4 - r^2)^2}{2}] \, drd\theta[/tex]
=> [tex]\int\limits^{\frac{\pi}{2}}_0 [(\frac{4}{3} r^{3} - \frac{r^{5}}{5})( cos \theta + sin \theta) + \frac{(4 - r^2)^3}{12}]^{2}_{0} \, d\theta[/tex]
Putting the limit and solving,
=> [tex]\int\limits^{\frac{\pi}{2}}_0 [(cos \theta + sin\theta )\frac{64}{15} + \frac{16}{3} \theta ]\, d\theta[/tex]
=> [tex][(sin\theta - cos\theta )\frac{64}{15} + \frac{16}{3} \theta ]^\frac{\pi}{2} _0[/tex]
Evaluating the limits,
[tex]\int\int\int\limits_E {x + y +z } \,dV = \frac{128}{15} + \frac{8\pi}{3}[/tex]
To know more about Cylindrical coordinates here
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