(a).

Observe the figure :

Let the point  be .

Let the point  be .

Distance .

Area of the triangle  is .

Where  is base and  is height.

Observe the figure :

Base is  and height is .

Area of the triangle  is .

(b).

Find the area of the shaded region in terms of .

Area of the sector part other than triangle in shaded are is calculated as follows.

.

Total area of the shaded region is .

Total area of the shaded region is .

(c).

Ratio  of  to that of the shaded region.

(d).

Evaluate .

.

Apply LHopitals Rule :

For  If  or  then .

Since limit is in indeterminate form of , again apply LHopitals Rule.

.

Since limit is in indeterminate form of , again apply LHopitals Rule.

.

(a).

Area of the triangle  is .

(b).

Total area of the shaded region is .

(c).

Ratio  of  to that of the shaded region is .

(d).

.