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Evaluate the line integral, where C is the given curve. C xy2 ds, C is the right half of the circle x2 + y2 = 4 oriented counterclockwise Incorrect: Your answer is incorrect.

Respuesta :

LammettHash

Parameterize the path by

[tex]\vec r(t)=(2\cos t,2\sin t)[/tex]

with [tex]-\dfrac\pi2\le t\le\dfrac\pi2[/tex]. Then the integral is

[tex]\displaystyle\int_Cxy^2\,\mathrm ds=\int_0^18\cos t\sin^2t\sqrt4\,\mathrm dt[/tex]

[tex]=\displaystyle16\int_0^1\cos t\sin^2t\,\mathrm dt=\boxed{\frac{32}3}[/tex]

okpalawalter8

The correct answer is [tex]C=\frac{32}{3}[/tex] as the Curve of the circle.

From the question we are told that:

Curve [tex]C =xy^2 ds[/tex]

Right half of the circle [tex]x^2 + y^2 = 4[/tex]

Since

Right half of the circle counterclockwise Incorrect:

Therefore

[tex]x=2cost[/tex]

[tex]y=2sint[/tex]

Giving that

[tex]ds=\sqrt{x^2(+1)^2+(y^2){+1}^2}dt[/tex]

[tex]ds=\sqrt{2^2(sin^2t+cos^2t}dt[/tex]

[tex]d_s=2dt[/tex]

Generally the curve

[tex]C=\int xy^2ds[/tex]

[tex]C=\int (2cost)(2sint)^2dt[/tex]

[tex]C=\frac{32}{3}[/tex]

In conclusion wit the Right half of the circle as  [tex]x^2 + y^2 = 4[/tex] it is deduced that the Curve counterclockwise is  [tex]C=\frac{32}{3}[/tex]

For more explanation on this visit

https://brainly.com/question/4698987?referrer=searchResults