find the area under the curve y=1/(6x^3)

Question

Find the area under the curve y=1/(6x^3) from x=1 to x=t and evaluate it for t=10, t=100. Then find the total area under this curve for x≥1.

 

(a) t=10

=____________

 

(b) t=100

=____________

 

(c) Total area

=____________

Answer 1

The anti-derivative of 1/(6 x³) is -(1/12)/ x²  

 

So the area under the curve is (1/12) ( 1 - 1/t² )

 

So  for t = 10    area =  (1/12) .99

     for  t = 100   area = (1/12) (.9999)

 

The entire area corresponds to t = ∞    ,  so area =  (1/12)