cal notes to sort {exm2 cal3} Polar integrals

We are given some upper half of a unit circle, tasked to find

• carteasian cords?

• polar cords?

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Q1) Write the plane $x+y+z=2$ as level curve of some function f. What is f and C?

Level curve is defined by some z=k, where we can easily set z equal to the rest of the equastion. So we simply let f(x) = z, hence $z=2-x-y$

Q2) Write ellipse with semimajor axis 4 and semi minor 2 as level curve of some func

• Ellipse: $x^2+{\left(\frac{y}{2}\right)}^2=1$
• Unit circle: $x^2+y^2=1$, d=2
• So have: $\boxed{f(x,y)=\frac{x^2}{4}+y^2}$

Q3) $4x^2+y^2\le 3$

• plot $x^2+y^2=3$
• Plot some $\nabla f$ vectors
• Shade in inequaltiy..