5.

(a) Prove that if E is a Jordan region whose topological boundary is a piecewise smooth curve oriented in the counterclockwise direction, then Area (E) = ~ f xdy-ydx.
2 JaE

(b) Find the area enclosed by the loop in the Folium of Descartes, i.e., by ( 3t 3t2)
¢(t) = 1 + t3' 1 + t3 ' t E [0, (0).

(c) Find an analogue of part

(a) for the volume of a Jordan region E in R 3 .

(d) Compute the volume of the torus with radii a > b (see Example 13.32).
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