
# 1 Question 1

Evaluate

\begin{equation*} \iint_\Omega x^2y \dd\mathrm{Vol}(x,y,z) \end{equation*}

where $$\Omega$$ is the volume obtained by interecting $$\Omega_1$$ $$\Omega_2$$ $$\Omega_3$$ where

• $$\Omega_1$$ are the points $$(x,y,z)$$ between $$y=\sqrt{x}$$ and $$y=0$$,
• $$\Omega_2$$ are the points $$(x,y,z)$$ with $$x$$ between $$1$$ and $$3$$.
• $$\Omega_3$$ are the points $$(x,y,z)$$ between the $$x,y$$ plane and the plane $$x+y-z=0.$$

## 1.1 Asessment

Understanding the shape of the domain: 30% Setting up the integral: 50% Computing the integral: 20% Expected time: 6min work + 2 min explanation

# 2 Question 3

Indentify all local and global extremal points of

\begin{equation*} (x^{2}+1) (y^{2}+1)-\frac{1}{2} x^{2}-(\frac{1}{2}) y^{2}-y^{4} \end{equation*}

on $$\R^2$$

If there are no global maximums or minimums, explain why.

## 2.1 Asessment

• Setting up : 50%
• Computation: 30%
• Explanation: 20%
• Expected time: 6min work + 2 min explanation

# 3 Question 4

Find the global and local maxima and the minima of the function

\begin{equation*} f(x,y)= x^2+y^2 \end{equation*}

on the boundary of the region $$\Omega$$ obtained by intersecting $$\Omega_1$$ and $$\Omega_2$$

• $$\Omega_1$$ is the region between the lines $$x=-1$$ and $$x=1$$
• $$\Omega_2$$ is the region between the lines $$y=0$$ and $$y^2=\frac{100}{1+x^2/100}$$.

If there are no global maximums or minimums, explain why.

## 3.1 Asessment

• Setting up : 50%
• Computation: 30%
• Explanation: 20%
• Expected time: 8min work + 2 min explanation