Mathematics: Evaluate the line integral c f · dr, where c is given by the vector function r(t). f(x, y, z) = (x + y2) i + xz j + (y + z) k, r(t) = t2 i + t3 j − 2t k, 0 ≤ t ≤ 2 $$ correct: your answer is correct.

Evaluate the line integral c f · dr, where c is given

Ответ:

woodpeckerdeejpe8wvh

07.01.2022

Mathematics

Given:

$\to \bold{\int_{c} \ F \cdot dx}$

Where

$\to \bold{F(x,y,z)=(x+y^2)i+xzj+(y+z)K}$ and C is given by the vector function $\bold{r(t)}$ .

$\to \bold{r(t)=t^2i+t^3j-2tk,\ \ 0\leq t\leq 2}\\\\\to \bold{dx=(2ti+3t^2j-2k)dt}\\\\$

therefore

$\to \bold{x=t^2}\\\\ \to \bold{y=t^3}\\\\ \to \bold{z=2t}\\\\\to \bold{F(r)=(t^2+(t^3)^2)i+t^2(-2t)j+(t^3-2t)k}\\\\$

$\bold{=(t^2+t^6)i-2t^3j+(t^3-2t)k}\\\\$

$\bold{\therefore}\\\\\bold{F(r).dx=((t^2+t^6)i-2t^3j+(t^3-2t)k).(2ti+3t^2j-2k)}\\\\$

$=\bold{((t^2+t^6)2t-2t^3(3t^2)+(t^3-2t)(-2)]dt}\\\\=\bold{(2t^3+2t^7-6t^5-2t^3+4t)dt}\\\\=\bold{(2t^7-6t^5+4t)dt}\\\\$

Calculating the line integral:

$\bold{\int_{c}\ F .dx=\int^{2}_{t=0} (2t^7-6t^5+4t)dt}\\\\$

$=\bold{[\frac{t^8}{4}-t^6+2t^2]^{2}_{0}}\\\\=\bold{\frac{2^8}{4}-2^6+2(2)^2}\\\\=\bold{\frac{256}{4}-64+2(4)}\\\\=\bold{64-64+8}\\\\=\bold{8}$

Therefore the final answer "8".

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Ответ:

sarn8899

07.01.2022

Mathematics

$\vec r(t)=t^2\,\vec\imath+t^3\,\vec\jmath-2t\,\vec k$

$\mathrm d\vec r=(2t\,\vec\imath+3t^2\,\vec\jmath-2\,\vec k)\,\mathrm dt$

$\vec f(x,y,z)=(x+y^2)\,\vec\imath+xz\,\vec\jmath+(y+z)\,\vec k$

$\vec f(x(t),y(t),z(t))=(t^2+t^6)\,\vec\imath-2t^3\,\vec\jmath+(t^3-2t)\,\vec k$

The line integral is

$\displaystyle\int_C\vec f\cdot\mathrm d\vec r=\int_0^2\bigg((t^2+t^6)\,\vec\imath-2t^3\,\vec\jmath+(t^3-2t)\,\vec k\bigg)\cdot(2t\,\vec\imath+3t^2\,\vec\jmath-2\,\vec k)\,\mathrm dt$

$=\displaystyle\int_0^2(2t^7-6t^5+4t)\,\mathrm dt=\boxed{8}$

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Ответ:

Hazy095

30.10.2020

Mathematics

500 is what you already have if it grows 2.75% a month you multiply it by the month and add it to 500

500+2.75%m

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Evaluate the line integral c f · dr, where c is given by the vector function r(t). f(x, y, z) = (x + y2) i + xz j + (y + z) k, r(t) = t2 i + t3 j − 2t k, 0 ≤ t ≤ 2 $$ correct: your answer is correct.

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