91Ó°ÊÓ

Verify Green's Theorem by evaluating both integrals $$ \int_{C} y^{2} d x+x^{2} d y=\int_{R} \int\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right) d A \text { for the given path. } $$ C: \text { square with vertices }(0,0),(4,0),(4,4),(0,4) $$

Sketch several representative vectors in the vector field. $$ \mathbf{F}(x, y)=x \mathbf{i}+y \mathbf{j} $$

Evaluate the line integral along the given path. \(\int_{C}\left(x^{2}+y^{2}+z^{2}\right) d s\) $$ \begin{array}{c}C: \mathbf{r}(t)=\sin t \mathbf{i}+\cos t \mathbf{j}+8 t \mathbf{k} \\ 0 \leq t \leq \pi / 2\end{array} $$

Use Green's Theorem to calculate the work done by the force \(F\) on a particle that is moving counterclockwise around the closed path \(C\). $$ \mathbf{F}(x, y)=\left(e^{x}-3 y\right) \mathbf{i}+\left(e^{y}+6 x\right) \mathbf{j}, C: r=2 \cos \theta $$

In Exercises 17-22, evaluate \(\int_{S} \int f(x, y, z) d S\). $$ f(x, y, z)=x^{2}+y^{2}+z^{2} ; S: z=x+2, \quad x^{2}+y^{2} \leq 1 $$

Use Green's Theorem to calculate the work done by the force \(F\) on a particle that is moving counterclockwise around the closed path \(C\). $$ \begin{aligned} &\mathbf{F}(x, y)=\left(x^{3 / 2}-3 y\right) \mathbf{i}+(6 x+5 \sqrt{y}) \mathbf{j}\\\ &C: \text { boundary of the triangle with vertices }(0,0),(5,0), \text { and }\\\ &(0,5) \end{aligned} $$

Find the total mass of two turns of a spring with density \(\rho\) in the shape of the circular helix \(\mathbf{r}(t)=3 \cos t \mathbf{i}+3 \sin t \mathbf{j}+2 t \mathbf{k}\) \(\rho(x, y, z)=z\)

Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. \(\int_{C} \frac{2 x}{\left(x^{2}+y^{2}\right)^{2}} d x+\frac{2 y}{\left(x^{2}+y^{2}\right)^{2}} d y\) \(C:\) circle \((x-4)^{2}+(y-5)^{2}=9\) clockwise from (7,5) to (1,5)

A stone weighing 1 pound is attached to the end of a two-foot string and is whirled horizontally with one end held fixed. It makes 1 revolution per second. Find the work done by the force \(\mathbf{F}\) that keeps the stone moving in a circular path. [Hint: Use Force \(=\) (mass)(centripetal acceleration).]

Define a surface integral of the scalar function \(f\) over a surface \(z=g(x, y)\). Explain how to evaluate the surface inteoral