Final answer:
To find the length L of the arc formed by the equation y = (1/8)(-4x² + 2ln(x)) from x = 3 to x = 7, we can use the formula for arc length. By finding the derivative of y with respect to x and plugging it into the arc length formula, we can evaluate the integral and find the length of the arc to be 2.5 units.
Step-by-step explanation:
To find the length L of the arc formed by the equation y = (1/8)(-4x² + 2ln(x)) from x = 3 to x = 7, we can use the formula for arc length:
L = ∫ [3 to 7] √(1 + (f'(x))²) dx
First, let's find the derivative of y with respect to x:
y' = (1/8)(-8x + 2/x) = -x + 1/4x
Now, we can plug in the derivative into the arc length formula:
L = ∫ [3 to 7] √(1 + (-x + 1/4x)²) dx
Simplifying the expression inside the square root:
L = ∫ [3 to 7] √(1 + x² - 1/2x + 1/16x²) dx
L = ∫ [3 to 7] √((17/16)x² - (1/2)x + 1) dx
Now, integrate the expression:
L = ∫ [3 to 7] √(17/16)x² - (1/2)x + 1 dx
L = ∫ [3 to 7] √(289/256)x² - (17/32)x + 17/16 dx
L = ∫ [3 to 7] (17/16)x - (17/32)x + 17/16 dx
L = ∫ [3 to 7] (1/16)x dx
Integrating gives:
L = (1/16) * (7² - 3²) = (1/16) * 40 = 2.5
Therefore, the length of the arc is 2.5 units.