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Exercise 1: Warming up: A Tripple Integral

Determine the tripple integral

$$\,\displaystyle{\int_1^2\,\int_1^2\,\int_1^2 \frac{xy}{z}\,\, \mathrm dx\,\mathrm dy\,\mathrm dz\,.}$$

Exercise 2: Parametrization of a Spatial Field

This exercise is solved by hand.

In a region $\,\Omega\,$ in $\,(x,y,z)$-space is given by the parametric representation

$$\quad \mr(u,v,w)=\big(\,\frac{1}{2}\,u^2-v^2\,,\,-uv\,,\,w\,\big)\,,\,\,\,u\in \left[\, 0,2\,\right]\,,\,\,v\in \left[\, 0,2\,\right]\,,\,\,w\in \left[\, 0,2\,\right]\,.$$
A

In $\,\Omega\,$ the point

$$\,P=\mr(1,1,1)\,$$

is given.

Drawn from $\,P\,$ the tangent vectors $\,\mr_u’(P)\,,\,\mr_v’(P)\,$ and $\mr_w’(P)\,$ span a parallel-epipedon. Determine the volume of this parallel-epipedon. Possibly illustrate using Maple.

B

Determine the Jacobi function corresponding to $\,\mr\,$.

C

Determine the volume of $\Omega\,$.

Exercise 3: The Unit Sphere

In the $\,(x,z)$-plane in space the profile curve $\,C\,$ is given by

$$(x,z)=\mathbf s(u)=(\,\sin(u),\cos(u)\,)\,,\,\,u\in [\,0\,,\pi\,]\,.$$
A

Determine a parametric representation for the surface $\mathcal S$ that appears by rotating $C\,$, viewed as a space curve, $\,2\pi\,$ about the $\,z$-axis.

B

Explain that every point $\,(x,y,z)\,$ on $\,\mathcal S\,$ fulfills the equation $\,x^2+y^2+z^2=1\,,$ and that $\,\mathcal S\,$ is the unit sphere with centre at the origin.

C

Determine the area of $\,\mathcal S\,$.

A profile field $\,M\,$ in the $\,(x,z)$-plane in space is given by the parametric representation

$$(x,y,z)=\mathbf s(u,v)=(u\sin(v),0,u\cos(v)\,)\,,\,\,u\in[\,0,1\,]\,,v\in[\,0,\pi\,]\,.$$
D

State a parametric representation K for the solid of revolution that appears when $\,M\,$ is rotated the angle $\,2\pi\,$ about the $\,z$-axis. Which geometrical object are we talking about?

E

Determine

$$\int_{K} (z+1)\,d\mu\,.$$

Exercise 4: Fields bounded by a Graph Surface

A function of two variables are given by

$$z=h\,(x,y)=x^2+y\,.$$
A

Given the rectangle

$$A=\,\left\{(x,y)\,|\,x\in[\,-1,1\,]\,\,\mathrm{and}\,\,y \in[\,0,2\,]\,\right\}$$

in the $\,(x,y)$-plane. we consider the spatial field $\,B\,$ between $\,A\,$ and the graph for $\,h\,$.

Find a parametric representation for $\,B\,$. Plot using the parametric representation the part of the graph for $h$ determined by B. Then determine the corresponding Jacobi function, and determine

$$\int_{B} x^2-y\,d\mu\,.$$

A region $\,C\,$ in the $\,(x,y)$-plane appears by shifting in parallel the unit circular disc

$$\,\left\{(x,y)\,|\,x^2+y^2\leq1\,\right\}\,$$

that lies in the first quadrant by the vector $\,(1,0)\,.$ We consider the spatial region $\,D\,$ between $\,C\,$ and the graph for $\,h\,$.

B

Find a parametric representation for $\,D\,$. Plot using the parametric representation the part of the graph for $\,h\,$ determined by $\,D\,$. Then determine the corresponding Jacobi function and the volume $\,D\,.$

Exercise 5: Surface of Revolution. Parametrization and Integration

In the $\,(x,z)$-plane in space is given the filled triangle T that has the vertices

$$(0,0,0)\,,\,\,(1,0,0)\,\,\,\mathrm{and}\,\,\,(0,0,1)\,.$$
A

Give a parametric representation for T.

B

State a parametric representation for the surface of revolution $\,\Omega\,$ that appears when T is rotated the angle $\,2\pi\,$ about the $\,z$-axis. Which geometrical object are we talking about?

C

Determine the volume of surface of revolution.

Opg 6: More about Spheres

Consider the sphere $\,F\,$ given by

$$\mr(u,v)=\big(\,R\sin(u)\cos(v)\,,R\sin(u)\sin(v)\,,R\cos(u)\,\big) \,,\,\,u\in [\,a\,,\,b\,]\,\,,\,\,v\in [\,c\,,\,d\,]\,.$$

Here it shall apply that $\,R\geq 0\,$, $0\leq a \leq b \leq \pi\,$ and $\,0\leq c\leq d\leq 2\pi\,.$

D

What is the meaning of the parameters $\,R,\,a,\,b,\,c\,$ and $d\,?$

E

Determine the area of $\,F\,$.

Exercise 7: Even More about Spheres

Consider the spatial region given by

$$\mr(u,v,w)=\big(\,u\sin(v)\cos(w)\,,u\sin(v)\sin(w)\,,u\cos(v)\,\big) \,,\,\,u\in [a\,,\,b]\,,\,\,v\in [c\,,\,d]\,,\,\,w\in [e\,,\,f]\,.$$
A

What is the meaning of the parameters?

Let $\,A\,$ be the region that is determined by the choice:

$$a=1\,,\,\,b=3\,,\,\,c=\frac{\pi}{4}\,\,,d=\frac{\pi}{3}\,,\,\,e=0\,,\,\,f=\frac{3\pi}{4}$$

and $\,B\,$ by the choice

$$ a=2\,,\,\,b=4\,,\,\,c=\frac{\pi}{4}\,\,,d=\frac{\pi}{2}\,,\,\,e=-\frac{\pi}{4}\,,\,\,f=\frac{\pi}{4}\,.$$
B

Describe in words each of the fields $\,A\,$, $\,B\,$ and $\,A\cap B\,,$, and determine their volume.

C

Find the integrals

$$\int_Ax\, d\Omega,\, \int_Bx\, d\Omega\,\mathrm{and}\, \int_{A\cap B}x\, d\Omega.$$