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Exercise 1: Integration by Parts and Substitution in Two Variables

A

Determine $\displaystyle{\int_0^{\frac{\pi}{2}}\left(\int_0^{\frac{\pi}{2}} u\cos(u+v)\,\mathrm du\right)\,\mathrm dv\,.}$

B

Determine $\displaystyle{\int_0^1\left(\int_0^1\, \frac{v}{(uv+1)^2}\,\mathrm du\right)\,\mathrm dv\,.}$

Exercise 2: Plane Integral with Parametrization

We shal determine the plane integral

$$\int_B \,2xy\,d\mu\, \quad\mathrm{where}\quad B=\left\lbrace (x,y)\,\vert\, 0\leq x\,, 0\leq y\,, x+y\leq 1\right\rbrace\,.$$

Follow the steps below.

C

First sketch the field $B.$ Then determine a parametric representation for $B\,$.

D

Determine the Jacobi function, that corresponds to this parametrization.

E

Now determine the wanted integral.

Exercise 3: Graph Surface, Parametrization and Integration

Intro: If a function of two variables over an axis parallel rectangle is given:

$$h(x,y)\,,\,\, x\in\left[ a,b\right] ,\,y\in\left[ c,d\right]\,,$$

it is easy to state a parametric representation for its graph:

$$ \begin{matr}{c}x\\\\y\\\\z\end{matr}=\mr(u,v)=\begin{matr}{c}u\\\\v\\\\h(u,v)\end{matr} \,,\,\, u\in\left[ a,b\right] ,\,v\in\left[ c,d\right]$$
A

Given the function $\,h(x,y)=\sqrt 3\,y\,$ and the set of points

$$\,M=\left\{(x,y)\,|\,x\in\left[ 0,1\right] ,\,y\in\left[ 0,2\right]\right\}\,.$$

Let $G$ denote the part of the graph for $h$ that lies vertically above $M\,.$ Determine a parametric representation for $G\,,$ and compute the surface integral

$$\int_G\,xyz\,\mathrm d\mu\,.$$

Exercise 4: Surface of Revolution, Parametrization and Intergral

Intro: A surface of revolution $F$ appears by rotating a profile curve given in the $(x,z)$-plane, about the $z$-axis. A parabola segment $K$ in the $(x,z)$-plane is given by the equation

$$z=\frac{x^2}4\,\,,\,\,x\in [\,0\,,\,2\,]\,.$$
A

Explain that $K$ considered to be a space curve can be written as a parametric representation

$$\mr(u)=(g(u),0,h(u))=\left(u,0,\frac{u^2}{4}\right), \quad\mathrm{where}\,u\in\left[\, 0,2\,\right] .$$

A surface of revolution $F$ appears from rotating the profile curve $K$ $2\pi$ about the $z$-axis.

B

Explain that $F$ can be described by the parametric representation \begin{align} \mr(u,v)&=\big(\,g(u)\cos(v),g(u)\sin(v),h(u)\,\big)\
&=\big(\,u\cos(v),u\sin(v),\frac{u^2}{4}\,\big)\,,\,\,u\in[\,0,2\,]\,\,,v\in[\,0,2\pi\,]\,. \end{align}

C

Now we introduce the function $f(x,y,z)=x^2+y^2\,,$ and we shall compute the surface integral

$$\int_F f\mathrm d\mu.$$

Exercise 5: Graph Surface, Parametrization and Integration

For a function of two variables

$$h(x,y)=2-x^2-y^2\,$$

we consider the following two graph surfaces: \begin{align} F&=\lbrace(x,y,z)\,\vert\,x\in \left[\,0\,,\,1\,\right]\,, y\in \left[\,0\,,\,2\,\right] \,\,\mathrm{and}\,\, z=h(x,y)\,\rbrace\,,\
G&=\lbrace(x,y,z)\,\vert\, x^2+y^2\leq 2\,\,\mathrm{og}\,\, z=h(x,y)\,\rbrace\,. \end{align}

A

Compute $\displaystyle{\int_F\sqrt{9-4z}\,d\mu}\,.$

B

Compute $\displaystyle{\int_G\sqrt{9-4z}\,d\mu}\,.$