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Exercise 1: Warming up: A Tripple Integral

A

Compute 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 Region

By hand.

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

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

Within $\,\Omega\,$ we are given the point

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

When drawn from $\,P\,$, the tangent vectors $\,\mr_u’(P)\,,\,\mr_v’(P)\,$ and $\mr_w’(P)\,$ span a parallelepiped.

Compute the volume of this parallelepiped.

B

Create an illustration using Maple of $\Omega$.

C

Determine the Jacobian that corresponds to $\,\mr\,$.

D

Compute the volume of $\Omega\,$.

Exercise 3: The Unit Sphere

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

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

Determine a parametric representation of the surface $\mathcal S$ that is formed when $C\,,$ viewed as a space curve, sweeps through an angle of $\,2\pi\,$ about the $\,z$ axis.

B

Show that every point $\,(x,y,z)\,$ on $\,\mathcal S\,$ fulfills the equation $\,x^2+y^2+z^2=1\,.$ Which geometric object are we talking about?

C

Compute the area of $\,\mathcal S\,$, first via well-known geometric formulas and then via integration.

A profile surface $\,M\,$ in the $\,(x,z)$ plane 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 of the solid of revolution $\mathcal K$ that is formed when $\,M\,$ is rotated by an angle of $\,2\pi\,$ about the $\,z$ axis. Which geometric object are we talking about?

E

Compute

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

Exercise 4: Regions Bounded by a Graph Surface

A function of two variables is 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 region $\,B\,$ that is located between $\,A\,$ and the graph of $\,h\,$.

State a parametric representation of $\,B\,$. Make a plot of $A$ and the “top” of $B$, meaning the part of $B$ that coincides with the graph of $h$. With these two plots you can “imagine” how the solid looks visually.

B

Determine the Jacobian that corresponds to your parametrization, and compute

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

A circular unit disc is described by

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

The part of this disc that is located in the first quadrant of the $(x,y)$ plane undergoes a parallel transformation (also known as a parallel shift) by the vector $\,(1,0)\,.$ The resulting surface is denoted $C$. We will now consider the spatial region $\,D\,$ between $\,C\,$ and the graph of $\,h\,$.

C

Choose a parametric representation of $\,D\,$. Plot the top and bottom surfaces of this parametrization (if you wish, then plot every surface wrapping the solid region $D$ to make a perfect plot of $D$). Compute the volume of $\,D\,.$

Exercise 5: Solid of Revolution. Parametrization and Integration

In the $\,(x,z)$ plane we are given a filled triangle $T$ that has the corners

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

Provide a parametric representation of $T$.

B

State a parametric representation of the solid of revolution $\,\Omega\,$ that is formed when $T$ is rotated by an angle of $\,2\pi\,$ about the $\,z$ axis. Which geometric object have you formed?

C

Compute the volume of $\Omega$.

Opg 6: More about Spheres

Consider the generalized spherical shell $\,F\,$ given by

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

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

A

Discuss with a fellow student the meaning of each of the values $\,R,\,a,\,b,\,c\,$ and $d\,.$

B

Compute 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

Discuss with a fellow student the meaning of the values $a,b,c,d,e$ and $f$.

B

If you want the spatial region to form a solid sphere, then how can you further detail the values $a,b,c,d,e$ and $f$?

Let $\,A\,$ be the region determined by the following choice of values:

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

and $\,B\,$ the region determined by the choice:

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

Describe in words each of the regions $\,A\,$, $\,B\,$ and $\,A\cap B\,.$ Compute all three volumes.

D

State the volume of $\,\mathrm{Vol}(A\cup B)\,$ from geometric consideration.

E

Compute the integrals

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