(1,2) , (7,10) Find the distance between the two points using integration.

 Given the equation of a line y = mx + b,
=> slope = dy/dx = m . Thus, the distance is:
L = int_a^b sqrt(1+(dy/dx)^2) dx where a<=x<=b
 
we know the two points (x_1,y_1)=(1,2)
(x_2,y_2)=(7,10)
m = (y_2- y_1)/(x_2-x_1) = (10-2)/(7-1) = 8/6=4/3
so now the length is L = int_1^7 sqrt(1+(4/3)^2) dx
 = int_1^7 sqrt(1+(16/9)) dx
 = int_1^7 sqrt(25/9) dx
= int_1^7 (5/3) dx
= (5/3) int_1^7 1 dx 
= (5/3) |_1^7 x
= (5/3)[7-1]
= (5/3)6 = 5*2 = 10
 
so the distance between the two points = 10