Item 15

Evaluate the surface integral. (give your answer correct to at least three decimal places.) s is the boundary of the region encl

telo118 [61]

Split up the surface $S$ into three main components $S_1,S_2,S_3$ , where

$S_1$ is the region in the plane $y=0$ bounded by $x^2+z^2=1$ ;

$S_2$ is the piece of the cylinder bounded between the two planes $y=0$ and $x+y=2$ ;

and $S_3$ is the part of the plane $x+y=2$ bounded by the cylinder $x^2+z^2=1$ .

These surfaces can be parameterized respectively by

$S_1:~\mathbf s_1(u,v)=\langle u\cos v,0,u\sin v\rangle$
where $0\le u\le1$ and $0\le v\le2\pi$ ,

$S_2:~\mathbf s_2(u,v)=\langle\cos v,u,\sin v\rangle$
where $0\le u\le2-\cos v$ and $0\le v\le2\pi$ ,

$S_3:~\mathbf s_3(u,v)=\langle u\cos v,2-u\cos v,u\sin v\rangle$
where $0\le u\le1$ and $0\le v\le2\pi$ .

The surface integral of a function $f(x,y,z)$ along a surface $R$ parameterized by $\mathbf r(u,v)$ is given to be

$\displaystyle\iint_Sf(x,y,z)\,\mathrm dS=\iint_Sf(\mathbf r(u,v))\left\|\frac{\partial\mathbf r(u,v)}{\partial u}\times\frac{\partial\mathbf r(u,v)}{\partial v}\right\|\,\mathrm du\,\mathrm dv$

Assuming we're just finding the area of the total surface $S$ , we take $f(x,y,z)=1$ , and split up the total surface integral into integrals along each component surface. We have

$\displaystyle\iint_{S_1}\mathrm dS=\int_{u=0}^{u=1}\int_{v=0}^{v=2\pi}u\,\mathrm dv\,\mathrm du$
$\displaystyle\iint_{S_1}\mathrm dS=\pi$

$\displaystyle\iint_{S_2}\mathrm dS=\int_{v=0}^{v=2\pi}\int_{u=0}^{u=2-u\cos v}\mathrm du\,\mathrm dv$
$\displaystyle\iint_{S_2}\mathrm dS=4\pi$

$\displaystyle\iint_{S_3}\mathrm dS=\int_{u=0}^{u=1}\int_{v=0}^{v=2\pi}\sqrt2u\,\mathrm dv\,\mathrm du$
$\displaystyle\iint_{S_3}\mathrm dS=\sqrt2\pi$

Therefore

$\displaystyle\iint_S\mathrm dS=\left\{\iint_{S_1}+\iint_{S_2}+\iint_{S_3}\right\}\mathrm dS=(5+\sqrt2)\pi\approx20.151$

3 0

3 years ago

How are same side interior angles related to one another?

tia_tia [17]

Answer:

The relation between the same side interior angles is determined by the same side interior angle theorem. The theorem for the "same side interior angle theorem" states: If a transversal intersects two parallel lines, each pair of same-side interior angles are supplementary (their sum is 180°).

8 0

3 years ago

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Find the measure of the remote exterior angle. <br> m∠x=(5n−19)°<br> m∠y=(n+7)°<br> m∠z=(144−6n)°

navik [9.2K]

Answer:

66°

Step-by-step explanation:

An exterior angle of a triangle is equal to the sum of the opposite interior angles.

x + y = z

5n − 19 + n + 7 = 144 − 6n

6n − 12 = 144 − 6n

12n = 156

n = 13

m∠z = (144−6n)°

m∠z = (144−6×13)°

m∠z = 66°

7 0

3 years ago

What values are needed to make each expression a perfect square trinomial?

Ivahew [28]

Answer:

4;

25

Step-by-step explanation:

x² + 2(x)(2) + 2²

x² + 4x + 4

x² - 2(x)(5) + 5²

x² - 10x + 25

3 0

3 years ago

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The method of completing the square was used to solve the equation 2x^2 - 12x + 6 =0 which equation is a correct step when using

trapecia [35]

The equation we are looking at would have to be divided by 2 first.
x²-6x+3
When looking at that, I can tell that 1 may be the possible answer since you would have to subtract 6 from both sides. That would leave a 3. Let's check.
(x-3)(x-3)
x²-6x+9=6
Now subtract 6 from both sides.
x²-6x+3=0
So, 1- (x-3)²=6 would be used.

6 0

3 years ago