Find the value of integral int_{C}left(x^{2}+y^{2}+zright) d s, where C is parmeterized by vec{r}(t)=langle 3 cos (4 t), 3 sin (4 t), 2 trangle for 0 leq t leq 4.

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Accepted Answer

Step:1 ：Let&#x27;s parameterize the given curve C by the vector function (vec{r}(t)=langle 3 cos (4 t), 3 sin (4 t), 2 trangle) for (0 leq t leq 4).Step:2 ：Substitute the parameterized values of x, y, and z into the function. So, x = 3*cos(4*t), y = 3*sin(4*t), and z = 2*t.Step:3 ：Substitute these values into the function to get (f = 2*t + 9*sin(4*t)^{2} + 9*cos(4*t)^{2}).Step:4 ：Integrate this function over the given limits of t from 0 to 4.Step:5 ：The value of the integral is 52.

Answer

Step: 1 ：Let&#x27;s parameterize the given curve C by the vector function (vec{r}(t)=langle 3 cos (4 t), 3 sin (4 t), 2 trangle) for (0 leq t leq 4).Step: 2 ：Substitute the parameterized values of x, y, and z into the function. So, x = 3*cos(4*t), y = 3*sin(4*t), and z = 2*t.Step: 3 ：Substitute these values into the function to get (f = 2*t + 9*sin(4*t)^{2} + 9*cos(4*t)^{2}).Step: 4 ：Integrate this function over the given limits of t from 0 to 4.Step: 5 ：The value of the integral is 52.

Find the value of integral $\int_{C} (x^{2} + y^{2} + z) d s$ , where $C$ is parmeterized by $\vec{r} (t) = ⟨ 3 \cos (4 t), 3 \sin (4 t), 2 t ⟩$ for $0 \leq t \leq 4$ .

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Popular Problems

Find the value of integral ∫C(x2+y2+z)ds, where C is parmeterized by r→(t)=⟨3cos⁡(4t),3sin⁡(4t),2t⟩ for 0≤t≤4.

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Popular Problems

Find the value of integral $\int_{C} (x^{2} + y^{2} + z) d s$ , where $C$ is parmeterized by $\vec{r} (t) = ⟨ 3 \cos (4 t), 3 \sin (4 t), 2 t ⟩$ for $0 \leq t \leq 4$ .