find the arc length of the graph of the function over the indicated interval. x^5/10+1/(6x^3) on the interval (3,6)
let y = x^5/10+1/(6x^3)
here is a picture of y = x^5/10+1/(6x^3) from 3 to 6
http://www.wolframalpha.com/input/?i=plot+y+%3D+x%5E5%2F10%2B1%2F%286x%5E3%29+from+3+to+6
arc length = ∫√( 1 + (dy/dx)^2) dx from 3 to 6
dy/dx = x^4/2 - 1/(2x^4)
(dy/dx)^2 = x^8/4 - 1/2 + 1/(4x^8)
= (1/4)x^8 - 1/2 + (1/4)x^-8
so arc
= ∫√(1 + (1/4)x^8 - 1/2 + (1/4)x^-8) dx from 3 to 6
nasty integral, I trusted Wolfram
http://www.wolframalpha.com/input/?i=integrate+%281+%2B+x%5E8%2F4+-+1%2F2+%2B+1%2F%284x%5E8%29+%29%5E%281%2F2%29+from+3+to+6
for appr 753.3 units of length
actually, the integral isn't so bad if you recognize that the integrand is just x^4 + 1/x^4
http://www.wolframalpha.com/input/?i=arc+length+of+x^5%2F10%2B1%2F%286x^3%29+from+3+to+6
yup, didn't see that perfect square
To find the arc length of a graph over a given interval, we need to integrate the square root of the sum of the squares of the derivative of the function with respect to x. In this case, we have the function f(x) = x^(5/10) + 1/(6x^3) and the interval (3,6).
1. Step 1: Find the derivative of the function f(x):
To find the derivative, we need to apply the power rule and the chain rule. The derivative of x^(5/10) is (5/10) * x^(-5/10-1), which simplifies to (1/2) * x^(-1/2). The derivative of 1/(6x^3) is -3/(6x^4), which simplifies to -1/(2x^4).
So, the derivative of the function f(x) is f'(x) = (1/2) * x^(-1/2) - 1/(2x^4).
2. Step 2: Find the integrand for the arc length formula:
To get the integrand, we need to take the square root of the sum of the squares of the x and y derivatives:
√(1 + [f'(x)]^2) = √(1 + [(1/2) * x^(-1/2) - 1/(2x^4)]^2).
Simplifying the expression inside the square root gives:
√(1 + (1/4) * x^(-1) - 1/(x^(3/2)) + 1/(4x^8)).
3. Step 3: Evaluate the integral:
Now, we need to integrate the above expression from x = 3 to x = 6:
Arc length = ∫[3 to 6] √(1 + (1/4) * x^(-1) - 1/(x^(3/2)) + 1/(4x^8)) dx.
Unfortunately, this integral does not have a simple closed-form solution. To evaluate it numerically, you can use numerical integration methods like the Trapezoidal Rule, Simpson's Rule, or a computer software tool like MATLAB or Wolfram Alpha.
Alternatively, you can use online numerical integration calculators that can provide an approximate value for the arc length based on the given function and interval.