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**lebdim** 1. The plane curve is defined with a polar equation $\displaystyle r = \cos(\phi)(4\sin^2(\phi) - 1)$. Find the area of this plane figure, bound by the curve.

2. We have the region $\displaystyle \mathbb{R}^3$ given by inequalities $\displaystyle x + y + z \leq 2, x \geq 0, y \geq 0, z \geq 0$. Find the volume of this region with the cut of slices in appropriate direction.

3. The curve $\displaystyle \mathcal{K}$ is given parametrically by $\displaystyle x = e^{-t}\cos(t), y = e^{-t}\sin(t)$.

a) Find the arc length of $\displaystyle \mathcal{K}$ on interval $\displaystyle t \in [0, \ln(2)]$.

b) Find the area of the region, described by a vector from $\displaystyle A(0,0,0)$ up to the curve $\displaystyle \mathcal{K}$ on the interval $\displaystyle t \in [0, \ln(2)]$.