Curves and line integrals

Verify that the curve

$$\gamma (t)=(t^2+1,2t^3),t\in [1,2],$$

is regular and compute the tangent vector at the point $\gamma (1)$.

Solution available to those who have unlocked the subject.

Determine whether the curve

$$\gamma (t)=(t^3,t^6),t\in [-1,1],$$

is regular or piecewise regular, and describe its trace.

Solution available to those who have unlocked the subject.

Write the Cartesian equation of the tangent line to the curve

$$\gamma (t)=(2\cos t,\sin t),t\in [0,2\pi ],$$

at the point $\gamma \left(\frac{\pi }{4}\right)$.

Solution available to those who have unlocked the subject.

Compute the length of the curve

$$\gamma (t)=(\cos t+t\sin t,\sin t-t\cos t),t\in [0,\pi ].$$

Solution available to those who have unlocked the subject.

Compute the length of the helix

$$\gamma (t)=\left(2\cos t,2\sin t,\sqrt{5}t\right),t\in [0,4\pi ].$$

Solution available to those who have unlocked the subject.

Compute the length of the graph of the function

$$f(x)=\frac{2}{3}{x}^{\frac{3}{2}},x\in [0,3].$$

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Determine the arc length function of the curve

$$\gamma (t)=(3t,4t),t\in [0,1],$$

and write the arc length parametrization.

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Compute the line integral

$${\int }_{\gamma }xyds,$$

where $\gamma$ is the segment from the origin to the point $(1,1)$.

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Compute the line integral

$${\int }_{\gamma }{z}^{2}ds,\gamma (t)=(\cos t,\sin t,t),t\in [0,\pi ].$$

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A wire lies along the segment

$$\gamma (t)=(3t,4t),t\in [0,1],$$

with linear density $\delta (x,y)=x$. Compute its mass.

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Determine the center of mass of the homogeneous wire lying along the quarter circle

$$\gamma (t)=(\cos t,\sin t),t\in \left[0,\frac{\pi }{2}\right].$$

Solution available to those who have unlocked the subject.

Compute the work of the field $F(x,y)=(y^2,x)$ along the curve

$$\gamma (t)=(t,t^2),t\in [0,1].$$

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Compute the work of the field $F(x,y)=(-y,x)$ along the circle with center at the origin and radius $2$, traversed counterclockwise.

Solution available to those who have unlocked the subject.