Mathematics: Evaluate the surface integral. s xz ds s is the boundary of the region enclosed by the cylinder y2 + z2 = 25 and the planes x = 0 and x + y = 10

Evaluate the surface integral. s xz ds s is the boundary

Ответ:

asseatingbandit

13.03.2022

Split up the surface $\mathcal S$ into three sub-surfaces $\mathcal S_1,\mathcal S_2,\mathcal S_3$ , where:

(1) $\mathcal S_1$ is the part of $\mathcal S$ in the plane x=0

parameterized by

$\mathbf s_1(u,v)=\langle0,u\cos v,u\sin v\rangle$

with $0\le u\le5$ and $0\le v\le2\pi$ ;

(2) $\mathcal S_2$ is the part of $\mathcal S$ in the plane x+y=10

parameterized by

$\mathbf s_2(u,v)=\langle10-u\cos v,u\cos v,u\sin v\rangle$

with $0\le u\le5$ and $0\le v\le2\pi$ ; and

(3) $\mathcal S_3$ is the part of $\mathcal S$ on the cylinder y^2+z^2=25

parameterized by

$\mathbf s_3(u,v)=\langle u,5\cos v,5\sin v\rangle$

with $0\le u\le10-5\cos v$ and $0\le v\le2\pi$ .

In each respective case, we get surface elements

$\mathrm dS_1=\|{{\mathbf s}_1}_u\times{{\mathbf s}_1}_v\|\,\mathrm du\,\mathrm dv=u\,\mathrm du\,\mathrm dv$
$\mathrm dS_2=\|{{\mathbf s}_2}_u\times{{\mathbf s}_2}_v\|\,\mathrm du\,\mathrm dv=\sqrt2u\,\mathrm du\,\mathrm dv$
$\mathrm dS_3=\|{{\mathbf s}_3}_u\times{{\mathbf s}_3}_v\|\,\mathrm du\,\mathrm dv=5\,\mathrm du\,\mathrm dv$

So the surface integral is

$\displaystyle\iint_{\mathcal S}xz\,\mathrm dS=\sum_{i=1}^3\iint_{\mathcal S_i}x(u,v)z(u,v)\,\mathrm dS_i$

The first integral is

$\displaystyle\iint_{\mathcal S_1}xz\,\mathrm dS_1=0$

The second is

$\displaystyle\iint_{\mathcal S_2}xz\,\mathrm dS_2=\int_{v=0}^{v=2\pi}\int_{u=0}^{u=5}(10-u\cos v)u\sin v(\sqrt2u)\,\mathrm du\,\mathrm dv=0$

The third is

$\displaystyle\iint_{\mathcal S_3}xz\,\mathrm dS_3=5\int_{v=0}^{v=2\pi}\int_{u=0}^{u=10-5\cos v}\mathrm du\,\mathrm dv=0$

and so

$\displaystyle\iint_{\mathcal S}xz\,\mathrm dS=0$

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Ответ:

Darkknighta26

09.06.2023

Mathematics

The total bill before the tip would be $ 25

Step-by-step explanation:

Let x be the amount of total bill,

∵ tip is 18% of the bill,

So, the amount of bill = 18% of x = 0.18x,

According to the question,

0.18x = 4.50

$\implies x =\frac{4.5}{0.18}=25$

Hence, the total bill before the tip would be $ 25.

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Evaluate the surface integral. s xz ds s is the boundary of the region enclosed by the cylinder y2 + z2 = 25 and the planes x = 0 and x + y = 10

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