Given curves: y = 2- |x -1| , y = sin x ; x = 0 and x =2

To find: The area bounded by the curves

Solution:

Step 1: Finding the points of intersection of given curves

To find the points of intersection, we equate the given curves:

2- |x -1| = sin x

We can solve this equation graphically or by inspection to get the points of intersection as (0, 0.841) and (2, 1.384)

Step 2: Sketching the curves

We can sketch the given curves as shown below:


Step 3: Finding the required area

The required area is the sum of the areas of the two regions enclosed by the curves and the x-axis.

Region 1: Bounded by y = 2- |x -1| , y = 0 and x = 0 to x = 2

This region is a trapezium with height 2 and bases given by the equations of the curves.

The area of this region can be found as:

A1 = ½ (2) [∫02 (2 - |x - 1|) dx]

A1 = ∫02 (2 - |x - 1|) dx

Now, we divide the integral into two parts:

∫02 (2 - |x - 1|) dx = ∫01 (2 - x + 1) dx + ∫12 (x - 1) dx

= [2x - x^2/2]1 + [x^2/2 - x + 1]2

= 1/2 + 1/2 + 1/2 = 1

Region 2: Bounded by y = sin x, y = 0 and x = 0 to x = 2

This region is a curvilinear triangle with base 2 and height given by sin x.

The area of this region can be found as:

A2 = ∫02 sin x dx

= [-cos x]02

= cos 0 - cos 2

= 1 - cos 2

Therefore, the total area enclosed by the given curves is:

A = A1 + A2

= 1 + (1 - cos 2)

= 2 - cos 2

Hence, the correct option is (a) 2 cos^2 1

A