The triple integral that is bounded by a paraboloid x = 4y2 4z2 given as
![16pi / 3](https://tex.z-dn.net/?f=16pi%20%2F%203)
Parabloid, x = 4y^2 + 4z^2
plane x = 4
x = 4y^2 + 4z^2
x = 4
4 = 4y^2 + 4z^2
4 = 4 (y^2 + z^2 )
1 = y^2 + z^2
from polar coordinates
y = r cos θ
z = r sin θ
r^2 = y^2 + z^2
<h3>limts of the integral</h3>
0 ≤ θ ≤ 2π
4r^2 ≤ x ≤ 4
0 ≤ r ≤ 1
![\int\limits\int\limits\int\limits {x} \, dV = \int\limits^1_0\int\limits^a_b\int\limits^c_d {x} \, dx ( rdrdz)](https://tex.z-dn.net/?f=%5Cint%5Climits%5Cint%5Climits%5Cint%5Climits%20%7Bx%7D%20%5C%2C%20dV%20%3D%20%5Cint%5Climits%5E1_0%5Cint%5Climits%5Ea_b%5Cint%5Climits%5Ec_d%20%7Bx%7D%20%5C%2C%20dx%20%28%20rdrdz%29)
where
a = 4
b = 4r^2
c = 2r
d = 0
The first integral using limits c and d gives:
![2pi\int\limits^1_0\int\limits^a_b {xr} \, dx](https://tex.z-dn.net/?f=2pi%5Cint%5Climits%5E1_0%5Cint%5Climits%5Ea_b%20%7Bxr%7D%20%5C%2C%20dx)
The second integral using limits a and b
![pi\int\limits^1_0 {16 } } \, rdr - pi\int\limits^1_0 {16r^{5} \, dx](https://tex.z-dn.net/?f=pi%5Cint%5Climits%5E1_0%20%7B16%20%7D%20%7D%20%5C%2C%20rdr%20-%20pi%5Cint%5Climits%5E1_0%20%7B16r%5E%7B5%7D%20%5C%2C%20dx)
![16pi\int\limits^1_0 { } } \, rdr - 16pi\int\limits^1_0 {r^{5} \, dx](https://tex.z-dn.net/?f=16pi%5Cint%5Climits%5E1_0%20%7B%20%7D%20%7D%20%5C%2C%20rdr%20-%2016pi%5Cint%5Climits%5E1_0%20%7Br%5E%7B5%7D%20%5C%2C%20dx)
![16pi\int\limits^1_0 { } } \, [r-r^{5}]dr](https://tex.z-dn.net/?f=16pi%5Cint%5Climits%5E1_0%20%7B%20%7D%20%7D%20%5C%2C%20%5Br-r%5E%7B5%7D%5Ddr)
The third integral using limits 1 and 0 gives:
![16pi / 3](https://tex.z-dn.net/?f=16pi%20%2F%203)
Read more on Triple integral here: brainly.com/question/27171802
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Answer:
-1
Step-by-step explanation:
Slope: ![m](https://tex.z-dn.net/?f=m)
when the given points are
and ![(x_2,y_2)](https://tex.z-dn.net/?f=%28x_2%2Cy_2%29)
Plug in two of the given points
![m=\frac{0-4}{0-(-4)} \\m=\frac{0-4}{0+4} \\m=\frac{-4}{4} \\m=-1](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B0-4%7D%7B0-%28-4%29%7D%20%5C%5Cm%3D%5Cfrac%7B0-4%7D%7B0%2B4%7D%20%5C%5Cm%3D%5Cfrac%7B-4%7D%7B4%7D%20%5C%5Cm%3D-1)
Therefore, the slope of the line is -1.
I hope this helps!
ANSWER and EXPLANATION
We want to order the functions from widest to narrowest:
![\begin{gathered} y=3x^2 \\ y=2x^2 \\ y=4x^2 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20y%3D3x%5E2%20%5C%5C%20y%3D2x%5E2%20%5C%5C%20y%3D4x%5E2%20%5Cend%7Bgathered%7D)
To do this, we have to plot the graphs of the functions by using a table of values.
Let us find the values of the functions for values of x = -2, 0, 2
For the first function:
![\begin{gathered} y=3x^2 \\ \Rightarrow x=-2: \\ y=3(-2)^2=3*4 \\ y=12 \\ \Rightarrow x=0: \\ y=3(0)^2=3*0 \\ y=0 \\ \Rightarrow x=2: \\ y=3(2)^2=3*4 \\ y=12 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20y%3D3x%5E2%20%5C%5C%20%5CRightarrow%20x%3D-2%3A%20%5C%5C%20y%3D3%28-2%29%5E2%3D3%2A4%20%5C%5C%20y%3D12%20%5C%5C%20%5CRightarrow%20x%3D0%3A%20%5C%5C%20y%3D3%280%29%5E2%3D3%2A0%20%5C%5C%20y%3D0%20%5C%5C%20%5CRightarrow%20x%3D2%3A%20%5C%5C%20y%3D3%282%29%5E2%3D3%2A4%20%5C%5C%20y%3D12%20%5Cend%7Bgathered%7D)
Hence, its table is:
For the second function:
![\begin{gathered} y=2x^2 \\ \Rightarrow x=-2: \\ y=2(-2)^2=2*4 \\ y=8 \\ \Rightarrow x=0: \\ y=2(0)^2=2*0 \\ y=0 \\ \Rightarrow x=2: \\ y=2(2)^2=2*4 \\ y=8 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20y%3D2x%5E2%20%5C%5C%20%5CRightarrow%20x%3D-2%3A%20%5C%5C%20y%3D2%28-2%29%5E2%3D2%2A4%20%5C%5C%20y%3D8%20%5C%5C%20%5CRightarrow%20x%3D0%3A%20%5C%5C%20y%3D2%280%29%5E2%3D2%2A0%20%5C%5C%20y%3D0%20%5C%5C%20%5CRightarrow%20x%3D2%3A%20%5C%5C%20y%3D2%282%29%5E2%3D2%2A4%20%5C%5C%20y%3D8%20%5Cend%7Bgathered%7D)
Hence, its table is:
For the third function:
![\begin{gathered} y=4x^2 \\ \Rightarrow x=-2: \\ y=4\left(-2\right)^2=4*4 \\ y=16 \\ \Rightarrow x=0: \\ y=4(0)^2=4*0 \\ y=0 \\ \Rightarrow x=2: \\ y=4(2)^2=4*4 \\ y=16 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20y%3D4x%5E2%20%5C%5C%20%5CRightarrow%20x%3D-2%3A%20%5C%5C%20y%3D4%5Cleft%28-2%5Cright%29%5E2%3D4%2A4%20%5C%5C%20y%3D16%20%5C%5C%20%5CRightarrow%20x%3D0%3A%20%5C%5C%20y%3D4%280%29%5E2%3D4%2A0%20%5C%5C%20y%3D0%20%5C%5C%20%5CRightarrow%20x%3D2%3A%20%5C%5C%20y%3D4%282%29%5E2%3D4%2A4%20%5C%5C%20y%3D16%20%5Cend%7Bgathered%7D)
Hence, its table is:
Now, let us plot the graphs of the functions:
Therefore, from the graph, we see that the order of the functions from widest to narrowest is: