Find the length of the curve 

yequals=three fifths x Superscript 5 divided by 3 Baseline minus three fourths x Superscript 1 divided by 3 Baseline plus 835x5/3−34x1/3+8 

for 

1less than or equals≤xless than or equals≤88.

To find the length of the curve given by the equation

\[
y = \frac{3}{5} x^{\frac{5}{3}} - \frac{3}{4} x^{\frac{1}{3}} + 8
\]

over the interval \(1 \leq x \leq 8\), we use the formula for the length of a curve:

\[
L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx
\]

### Step 1: Compute the derivative \(\frac{dy}{dx}\)

Given:

\[
y = \frac{3}{5} x^{\frac{5}{3}} - \frac{3}{4} x^{\frac{1}{3}} + 8
\]

Differentiate \(y\) with respect to \(x\):

\[
\frac{dy}{dx} = \frac{d}{dx} \left(\frac{3}{5} x^{\frac{5}{3}}\right) - \frac{d}{dx} \left(\frac{3}{4} x^{\frac{1}{3}}\right) + \frac{d}{dx}(8)
\]

The derivative is:

\[
\frac{dy}{dx} = \frac{3}{5} \cdot \frac{5}{3} x^{\frac{5}{3} - 1} - \frac{3}{4} \cdot \frac{1}{3} x^{\frac{1}{3} - 1} + 0
\]

Simplifying:

\[
\frac{dy}{dx} = x^{\frac{2}{3}} - \frac{1}{4} x^{-\frac{2}{3}}
\]

### Step 2: Set up the integral

The length of the curve is:

\[
L = \int_{1}^{8} \sqrt{1 + \left(x^{\frac{2}{3}} - \frac{1}{4} x^{-\frac{2}{3}}\right)^2} \, dx
\]

### Step 3: Evaluate the integral

This integral can be evaluated numerically as it does not have a straightforward analytical solution. Let's calculate it.

I'll proceed to evaluate this integral.It looks like evaluating the integral analytically took too long, so let's approach it numerically instead. I'll calculate the length of the curve using a numerical integration method.The length of the curve over the interval \(1 \leq x \leq 8\) is approximately \(19.35\) units.

Would you like more details on the process or have any questions?

Here are some related questions you might find interesting:
1. How does the length of a curve change if you alter the coefficients of the function?
2. What would be the length of the curve over a different interval, such as \(2 \leq x \leq 6\)?
3. How does the complexity of the integrand affect the difficulty of evaluating the integral?
4. Can the curve length formula be applied to a parametric curve? If so, how?
5. What happens to the curve length if the function has discontinuities?

**Tip:** For complex integrals, numerical methods like Simpson's rule or Gaussian quadrature can provide accurate results efficiently.

Find the length of the curve y = 3/5 * x^(5/3) - 3/4 * x^(1/3) + 8 for 1 ≤ x ≤ 8.

Explore the process of finding the length of the curve defined by y = 3/5 * x^(5/3) - 3/4 * x^(1/3) + 8 over the interval 1 ≤ x ≤ 8. This problem involves key calculus concepts such as differentiation and integration, and is suitable for students in Grades 11-12.

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