With respect to the innermost integral: Use the method of substitution by setting the inner function to $g(u)=1+uv$ and the outer function to $f(u)=\displaystyle{\frac 1u}\,.$
answer
$$\int_B \frac{y}{1+xy} \;d\mu=2\ln 2-1.$$
Exercise 2: Parametrization and the Plane Integral. By Hand
In the $(x,y)$-plane the point $\displaystyle{P_0=\left(2\,,1\right)\,}$ is given together with the set of points
Make a preliminary sketch of $B\,$ and state a parametric representation $\mr(u,v)$ for $B\,$ with suitable intervals for $u$ and $v\,.$ Determine two numbers $u_0$ and $v_0$ such that $\mr(u_0,v_0)=P_0\,.$
B
Make an illustration of $B$ using Maple where you from $P_0$ draw the tangent vectors $\mr’_u(u_0,v_0)$ and $\mr’_v(u_0,v_0)\,.$ Determine the area of the parallelogram spanned by the tangent vectors.
C
Determine the Jacobi function corresponding to $\mr(u,v)$, and compute the plane integral
The parametrization can e.g. be given by $x=u$ and $\displaystyle{y=\frac 12\,vu^2}\,$ where $\displaystyle{u\in\left[ \frac 32\,,\,\frac 52\right]}\,$ og $v\in\left[ 0,1\right] $. The corresponding Jacobi function is then $\displaystyle{\frac 12\,u^2}\,.$
A function $\,f:\reel^2\rightarrow \reel\,$ is given by
$$\,\displaystyle{f(x,y)=x^2-y^2}\,.$$
For a given point in the $\,(x,y)$-plane $\,\varrho\,$ denotes the point’s absolute value (the distance from the point to the origin). Similarly, $\,\varphi\,$ denotes the argument of the point (the angle between the $x$-axis and the position vector of the point, computed with a sign in the counter-clockwise direction).
A set of points $B$ is in polar coordinates described by
where $\,a\,$ is an arbitrary positive real number.
A
Make a sketch of $\,B\,$, and determine the area of $\,B\,$ both with respect to integration and elementary geometric consideration.
hint
You can use this parametric representation of $\,B\,$:
$(x,y)=\mathbf r(u,v)=(\,u\,\cos(v)\,,\,u\,\sin(v)\,)\,$ for $-\frac{\pi}{4} \leq v \leq \frac{\pi}{2}\,$ og $\,0\leq u \leq a\,.$
answer
The area is $\frac 38 a^2\pi\,$.
B
Determine the plane integral $\displaystyle{\int_B f(x,y) \;d\mu}\,.$
answer
$$\int_B (x^2-y^2) \;d\mu=\frac{a^4}{8}.$$
Exercise 4: Parametric Surfaces
The exercise must be solved by hand.
A surface $\,F_{\mr}\,$ is given by the parametric representation
A point on the surface is given by $\,P_0=(0,1,\frac{\pi}{2})\,.$
Determine two numbers $\,u_0\,$ and $\,v_0\,$ such that $\,\mr(u_0,v_0)=P_0\,.$
Drawn from $\,P_0\,$ the tangent vectors $\,\mr’_u(u_0,v_0)\,$ and $\,\mr’_v(u_0,v_0)\,$ span a parallelogram $\,\mathcal P\,.$
State a parametric representation for $\,\mathcal P\,.$
answer
$\mr(1,\frac{\pi}{2})=(0,1,\frac{\pi}{2}).$
B
Make an Maple-illustration that contains:
1) $F_{\mr}\,$
2) $\mr’_u(u_0,v_0)\,$ and $\,\mr’_v(u_0,v_0)\,$
3) The normal vector $\,\mathbf n(u_0,v_0)=\mr’_u(u_0,v_0)\times \mr’_v(u_0,v_0)\,$
4) $\,\mathcal P\,.$
C
Determine the area of $\,\mathcal P\,.$
hint
The area is found as the length of the normal vector.
answer
$\sqrt 2\,.$
D
Find the Jacobi function corresponding to $\mr(u,v)$, And determine the area of $F_{\mr}\,.$
hint
$\mathbf n(u,v)=(\sin v, -\cos v,u).$
hint
The area is found by integrating the Jacobi-funktion =$\,|\mathbf n(u,v)\,|\,$ over the parametric field in the $(u,v)$-plane. Find an indefinite integral using Maple.
answer
$(2\sqrt 5-\ln(\sqrt 5-2))\pi \approx 18.6\,.$
Exercise 5: Cylindrical Surface. Parametrization of an Integral
A cylindrical surface is a surface that is vertically perpendicular to a so-called directrix in the $(x,y)$-plane. For the cylindrical surface to be well defined, a $z$-interval must be given for all points $(x,y)$ on the directrix. In this exercise we consider these two cylindrical surfaces:
