(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).

.