1 pt) Find the length of the curve defined by
y=1/8(x^2−8ln(x)) from x=3 to x=5
To find the length of a curve, we can use the arc length formula. The formula for the arc length of a curve defined by the equation y = f(x) from x = a to x = b is:
L = ∫[a,b]√(1 + (f'(x))^2) dx
First, let's find the derivative of the function y = 1/8(x^2 - 8ln(x)). Differentiating, we get:
y' = 1/8 * (2x - 8/x)
Simplifying further, we get:
y' = (1/4)x - 1
Now, let's find the integral of √(1 + (f'(x))^2) using the given values of a = 3 and b = 5:
L = ∫[3,5]√(1 + ((1/4)x - 1)^2) dx
This integral can be solved using various numerical or approximation methods, such as numerical integration techniques or software. One common method is to use a numerical integration method called the trapezoidal rule.
Using the trapezoidal rule, we can approximate the integral as follows:
L ≈ Δx/2 * (f(x_1) + 2f(x_2) + 2f(x_3) + ... + 2f(x_(n-1)) + f(x_n))
where Δx = (b - a) / n, n is the number of intervals, and x_1, x_2, ..., x_n are the x-values corresponding to the sampled points.
Let's choose a value for n and calculate the length approximately using the trapezoidal rule. For simplicity, let's choose n = 4.
Δx = (5 - 3) / 4 = 1/2
L ≈ (1/2) / 2 * (y(3) + 2y(3.5) + 2y(4) + 2y(4.5) + y(5))
Now, substitute the x-values into the equation y = 1/8(x^2 - 8ln(x)):
L ≈ (1/4) * ( (3^2 - 8ln(3)) + 2(3.5^2 - 8ln(3.5)) + 2(4^2 - 8ln(4)) + 2(4.5^2 - 8ln(4.5)) + (5^2 - 8ln(5)) )
Evaluating this expression will give us an approximate length of the curve defined by the given equation.