Respuesta :

LammettHash
Parameterize [tex]C[/tex] by

[tex]\mathbf r(t)=(x(t),y(t),z(t))=(1-t)(0,5,2)+t(2,7,8)=(2t,5+2t,2+6t)[/tex]
[tex]\implies\mathrm ds=\sqrt{\left(\dfrac{\mathrm dx}{\mathrm dt}\right)^2+\left(\dfrac{\mathrm dy}{\mathrm dt}\right)^2+\left(\dfrac{\mathrm dz}{\mathrm dt}\right)^2}\,\mathrm dt=\sqrt{44}\,\mathrm dt[/tex]

So the line integral is equivalent to

[tex]\displaystyle\int_Cx^2z\,\mathrm ds=\sqrt{44}\int_{t=0}^{t=1}(2t)^2(2+6t)\,\mathrm dt=\frac{52\sqrt{11}}3[/tex]
facundo3141592

We want to evaluate the line integral: x^2*zds on the line segment that goes from (0,5,2) to (2,7,8), we will get that the line integral is equal to 24.

So we just need to solve a triple integral, notice that we don't have a dependence on y, then we write the points just as (0, 2) and (2, 8)

Then we can find a linear relation between x and z, such that the slope of that line is:

[tex]a = \frac{8 - 2}{2 - 0} = 3[/tex]

And when x = 0, we have z = 2.

Then the linear equation is:

z = 3*x + 2

Then the line integral is just:

[tex]\int\limits^2_0 {x^2(3*x + 2)} \, dx *\int\limits^7_5 {} \, dy[/tex]

Where the second integral is just equal to 2, so we have:

[tex]2\int\limits^2_0 {x^2(3*x + 2)} \, dx \\\\2*[(2^3 - 0^3) + (2^2 - 0^2)] = 24[/tex]

If you want to learn more, you can read:

https://brainly.com/question/12440198

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