Question

Find the average value gave of the function g on the given interval. g(x) = 5x2 1 + x3, [0, 2]

Answer 1

The average value of over an interval [a, b Fav = te s fresda function ] is given flae) by - we have function 1 glas= 5x² [itx over [0, 2] = [a, b] of Average dar - goods over [0, 2] is - 1 aus de Q-0 gav = 1 1 58² Jitxa de o Let Hx²= t - ③ => Differentiate W.Yt 'x'. - 3x²= dt » 3x² dx = dt > x*dx = of 3

€ Also from h we have - When x=0, t= It 2013 t=1 & When He te Itajo t = 18 t = 9 -6 3 olor alus Using ②, ③, ④, ③ in 1 we get - Jav a á jstt. Lava & I've at gawa sa s tle at Lav ega leta) 11ftat en Lav = B x [1997–1998] dav = [1899-1] How = = [(32-1] als als

gav = ( 27-1] gav = 5 x 26 lav = 8 van dan Answel. .

Answer 2

SOLUTION :

Average value of the given function in interval 0 to 2 :

   2

= ∫  f(x) dx  / (2 - 0)

   0

            2

=   1/2 ∫  5x^2 sqrt(1 + x^3) dx

           0

Let 1 + x^3 = t

=> 3x^2 dx = dt

=> 5 x^2 dx = 5dt/3 

At x = 0, t = 1 and at x = 2, t = 9

So, average value :

          9                                                      9

= 1/2 ∫  5/3 sqrt(t) dt = 5/6 * 2/3 [ t^(3/2) ]                            

         1                                                       1

= 5/9 [9^(3/2) - 1^(3/2)]

= 5/9 * 26

= 130/9 = 14.44 (ANSWER)