(a)

The polar curve is , .

The polar curve is rotated about the line .

If is rotated by an angle , then the graph of curve will become .

Thus, when is rotated by an angle , the graph will become  .

From the result of 55(a),

is the surface area of the curve about polar axis.

Surface area of the resulting area after rotating the curve about the line .

Replace with in the surface area formula.

.

is the surface area of the curve about .

(b)

The polar curve is .

Differentiate with respect to  on each side.

.

Graph:

Graph the function .

Since the curve is rotated about line so find the area right part of the area, which is total surface area.

Substitute corresponding valuees in expression.

sq.units.

(a)

(b) sq.units.