Let C be the portion of the curve y =8�ãx between (1,8) and (25, 40). Find�çC 2yds.
To find the length of a curve, we can use the formula for arc length:
L = ∫(a to b) √(1 + (dy/dx)²) dx
Given the curve y = 8x, we need to find dy/dx and apply the formula to find the length of the curve.
First, let's find dy/dx by taking the derivative of y with respect to x:
dy/dx = d/dx (8x) = 8
Now we can substitute dy/dx into the formula for arc length:
L = ∫(a to b) √(1 + (8)²) dx
The integral represents the area under the curve between the limits a and b.
Given that the curve is y = 8x, we will integrate with respect to x between the x-values 1 and 25:
L = ∫(1 to 25) √(1 + 8²) dx
L = ∫(1 to 25) √(1 + 64) dx
L = ∫(1 to 25) √65 dx
Now we can integrate to find the length of the curve:
L = ∫(1 to 25) (√65) dx
L = (√65) ∫(1 to 25) dx
L = (√65)[x] (evaluated from 1 to 25)
L = (√65)(25 - 1)
L = (√65)(24)
L = 24√65
Therefore, the length of the curve C is 24√65 units.