Consider the plane ax+by+cz=d, where a, b, c, and d are all positive. Use previous problem to show that the volume of the tetrahedron cut from the first octant by this plane is $d D /\left(3 \sqrt{a^2+b^2+c^2}\right)$, where D is the area of that part of the plane in the first octant.

Let $\mathbf{F}=2 z \mathbf{i}+2 y \mathbf{k}$, and let $\partial S$ be the intersection of the cylinder $x^2+y^2=a y$ with the hemisphere $z=\sqrt{a^2-x^2-y^2}$, $a>0$. Assuming distances in meters and force in newtons, find the work done by the force $\mathbf{F}$ in moving an object around $\partial S$ in the counterclockwise direction as viewed from above.

In given problem, show that the given line integral is independent of path (use Theorem $\mathrm{C}$ ) and then evaluate the integral (either by choosing a convenient path or, if you prefer, by finding a potential function f and applying Theorem A).

$$\int_{(0,0,0)}^{(1,1,1)}\left(6 x y^3+2 z^2\right) d x+9 x^2 y^2 d y+(4 x z+1) d z$$

Hint: Try the path consisting of line segments from $(0,0,0)$ to $(1,0,0)$ to $(1,1,0)$ to $(1,1,1)$.

Let $\mathbf{F}=2 \mathbf{i}+x z \mathbf{j}+z^3 \mathbf{k}$ and $\partial S$ be the boundary of the surface $z=x^2 y^2, x^2+y^2 \leq a^2$, oriented counterclockwise as viewed from above. Evaluate $\oint_{\partial S} \mathbf{F} \cdot \mathbf{T} d s$.

Use the result of previous problem to verify the formula for the volume of a right circular cylinder of height h and radius a.

In given problem, show that the given line integral is independent of path (use Theorem $\mathrm{C}$ ) and then evaluate the integral (either by choosing a convenient path or, if you prefer, by finding a potential function f and applying Theorem A).

$$\int_{(-1,1)}^{(4,2)}\left(y-\frac{1}{x^2}\right) d x+\left(x-\frac{1}{y^2}\right) d y$$

Find the center of mass of the homogeneous triangle with vertices (a, 0,0),(0, b, 0), and (0,0, c), where a,b, and c are all positive.

If $\mathbf{F}=\left(x^2+y^2\right) \mathbf{i}+2 x y \mathbf{j}$, calculate the circulation of $\mathbf{F}$ around C of given problem; that is, calculate $\oint_C \mathbf{F} \cdot \mathbf{T} d s$.

In given problem, evaluate each line integral.

Same integral as in previous problem; C is the line segment from (0,0,0) to (2,3,4).

In given problem, find div $\mathbf{F}$ and curl $\mathbf{F}$.

$$\mathbf{F}(x, y, z)=\cos x \mathbf{i}+\sin y \mathbf{j}+3 \mathbf{k}$$

In given problem, find div $\mathbf{F}$ and curl $\mathbf{F}$.

$$\mathbf{F}(x, y, z)=y z \mathbf{i}+x z \mathbf{j}+x y \mathbf{k}$$

Find the work done by $\mathbf{F}=\left(x^2+y^2\right) \mathbf{i}-2 x y \mathbf{j}$ in moving a body counterclockwise around the curve C of given problem.

Let C be the circle that is the intersection of the plane $a x+b y+z=0(a \geq 0, b \geq 0)$ and the sphere $x^2+y^2+z^2=9$. For $\mathbf{F}=y \mathbf{i}-x \mathbf{j}+3 y \mathbf{k}$, evaluate

$$\oint_C \mathbf{F} \cdot \mathbf{T} d s$$

Hint: Use Stokes's Theorem.

Let $\mathbf{F}=2 \mathbf{i}+x z \mathbf{j}+z^3 \mathbf{k}$ and $\partial S$ be the boundary of the surface $z=x y^2, 0 \leq x \leq 1,0 \leq y \leq 1$, oriented counterclockwise as viewed from above. Evaluate $\oint_{a S} \mathbf{F} \cdot \mathbf{T} d s$.