Question

find the area of the shaded region

7. Find the area of the shaded region. f(x) = fx 1 x g(x) = - i o iż 3x

Answer 1

Area = f S dady - da dy -|' 13-24 - y2)dy = 3-1 - $ = 3-4 C

Answer 2

SOLUTION :

f(x) = √x and g(x) = 3/2 - x/2 

Point of intersection :

f(x) = g(x)

=> √x = 3/2 - x/2 

=> 2√x = 3 - x

Squaring

=> 4x = 9 - 6x + x^2

=> x^2 - 10x + 9 = 0

=> (x - 1)(x - 9) = 0

=> x = 1, 9 

X = 9 is irrelevant here,

So, x = 1 .

As per given figure :

             

Shaded Area

   

     1                3

 =  ∫  f(x) dx. + ∫   g(x) dx

     0                1

    1                      3

=  ∫  x^(1/2) dx  + ∫  (3/2 - x/2) dx

   0                       1

                       1                         3

= [2/3 x^(3/2)]  + [3/2 x - x^2/4}

                       0                         1

= 2/3 + [(3/2 (3) - 3^2 /4) - (3/2 (1) - 1^2/4)]

= 2/3 + [9/4 - 5/4]

= 2/3 + 1

= 5/3 ( ANSWER).