Mathematics, 20.07.2019 22:10 organicmemez

Evaluate x2 dv, e where e is the solid that lies within the cylinder x2 + y2 = 4, above the plane z = 0, and below the cone z2 = 9x2 + 9y2.

Answers: 1

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Answer from: ayoismeisalex

Following are the solution to the given equation:

Given

$\int \int \int x^2 dV$ wherein E is indeed the solid that exists inside the cylinders x^2+y^2=4 above surface z = 0 , that is below the cones z^2 = 9x^2+9y^2 E's precession through into xy-plane is $x^2+y'^2 \leq 4 \to r \leq 2$ with $\theta$ in $[0,2 \pi ] \\\\$

$\int \int \int x^2 \ dV =\int^{2\pi}_{0} \int^{2}_{0}\int^{3r}_{z= 0} r^2 \cos^2 \theta r \ dz \ dr \ d\theta \\\\$

$=\int^{2\pi}_{0} \int^{2}_{0}\int^{3r}_{z= 0} r^3 \cos^2 \theta \ dz \ dr \ d\theta \\\\= \int^{2\pi}_{0} \int^{2}_{0} 3r^4 \cos^2 \theta \ dr \ d\theta \\\\= 3 \int^{2\pi}_{0} \frac{1}{2} \frac{2^5}{5}(1+\cos\ 2 \theta) \ d\theta \\\\= \frac{48}{5}[ (\theta +\frac{1}{2}\sin 2 \theta) ]^{2\pi}_{0} \\\\= \frac{96\pi}{5}$

Therefore, the final answer is " $\frac{96\pi}{5}$ " .

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Answer from: Baby010391

Convert to cylindrical coordinates, setting

$\begin{cases}x=r\cos t\\y=r\sin t\\z=z\end{cases]$

Then

$\displaystyle\iiint_Ex^2\,\mathrm dV=\int_{t=0}^{t=2\pi}\int_{r=0}^{r=2}\int_{z=0}^{z=3r}r^3\cos^2t\,\mathrm dz\,\mathrm dr\,\mathrm dt=\frac{96\pi}5$

Answer from: Quest

a. we have two

lines: y = 4-x and

y = 8-x^-1

given two simultaneous equations that are both to be

true, then the solution is the points where the lines cross. the intersection is

where the two equations are equal. therefore the solution that works for both

equations is when

4-x = 8-x^-1

this is where the two lines will cross and that is the

common point that satisfies both equations.

b. 4-x = 8-x^-1

x 4-x 8-x^-1

-3 7

8.33

-2 6 8.5

-1 5 9

0 4

1 3

2 2

7.5

3 1 7.67

the table shows that none of the x values from -3 to 3 is

the solution because in no case does

4-x = 8-x^-1

to find the solution we need to rearrange the equation to

find for x:

4-x = 8-x^-1

multiply both sides with x:

4x-x^2 = 8x-1

x^2+4x-1=0

x= -4.236, 0.236

therefore there are two points that satisfies the

equation.

find y:

x=-4.236

y = 4-x = 4 – (-4.236)

= 8.236

y = 8-x^-1 = .236)^-1

= 8.236

x=0.236

y = 4-x = 4 – (0.236) = 3.764

y = 8-x^-1 = 8-(0.236)^-1

= 3.764

thus the two lines cross at 2 points:

(-4.236, 8.236) & (0.236, 3.764)

c. to solve

graphically the equation 4-x = 8-x^-1

we would graph both lines: y = 4-x and y

= 8-x^-1

the point on the graph where the lines cross is the

solution to the system of equations.

just graph the points on part b on a cartesian coordinate

system and extend the two lines. the

solution is, as stated, the point where the two lines cross on the graph.

step-by-step explanation:

mark brainliest!

Answer from: Quest

It is b as the correct answer is 0.088, you are able to work these out on a calculator or without a calculator if you type in the exact sum that is given for harder questions like these!

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