To find the arc length, we can use the formula:
\[L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\]
In this case, \(y = \frac{8x^3}{3} + \frac{1}{32x}\) on the interval \([1, 5]\).
First, let's find \(\frac{dy}{dx}\):
\[\frac{dy}{dx} = 8x^2 - \frac{1}{32x^2}\]
Now we can substitute this into the formula for arc length:
\[L = \int_1^5 \sqrt{1 + \left(8x^2 - \frac{1}{32x^2}\right)^2} \, dx\]
Simplifying the expression inside the square root gives:
\[L = \int_1^5 \sqrt{1 + 64x^4 - \frac{1}{16x^4}} \, dx\]
Now we can integrate:
\[L = \int_1^5 \sqrt{\frac{1024x^4 + 16x^8 - 1}{16x^4}} \, dx\]
To simplify this expression further, we can split it into two separate integrals:
\[L = \int_1^5 \frac{\sqrt{1024x^4 + 16x^8 - 1}}{4x^2} \, dx\]
\[L = \frac{1}{4} \int_1^5 \frac{\sqrt{1024x^4 + 16x^8 - 1}}{x^2} \, dx\]
Unfortunately, this integral does not have a simple closed-form solution. You would need to numerically approximate it using numerical integration methods such as Simpson's rule or numerical software.
Find the arc length of the following curve on the given interval by integrating with respect to x.
y=(8x^3 /3) + (1/32x) on [1,5]
An integral that gives the arc length is:
The arc length is:
1 answer