6. Find the length of the portion of the helix x(t) = cost, y(t) = sin t, z(t) = 2t lying between the planes z = 2 and z = 4.

Key words: a path integral, the length of a curve as an integral, parametrization of a curve.

7. A cardioid is the curve defined in polar coordinates by the equation r = 1 + sin . Find the length of a cardioid.

Key words: a path integral, the length of a curve as an integral, parametrization of a curve.

8. A ring made of thin wire has the shape of the circle x2 + y2 = 4x. The linear density of the wire at point (x, y) is equal to (x, y) = px2 + y2 units. Find the total mass ofthe ring.

Key words: a path integral, parametrization, polar coordinates.