y = 
2
3

x2 + 1
	3⁄2
 
Find the arc length of the graph of the function over the indicated interval. btween 0,3

To find the arc length of the graph of the function \( y = \frac{2}{3} x^2 + \frac{1}{2} \) over the interval \([0, 3]\), we can use the arc length formula:

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

### Step 1: Differentiate the function
The given function is:
\[
y = \frac{2}{3} x^2 + \frac{1}{2}
\]

Now, differentiate \( y \) with respect to \( x \):
\[
\frac{dy}{dx} = \frac{d}{dx} \left( \frac{2}{3} x^2 + \frac{1}{2} \right) = \frac{4}{3} x
\]

### Step 2: Square the derivative
Next, we square \( \frac{dy}{dx} \):
\[
\left( \frac{dy}{dx} \right)^2 = \left( \frac{4}{3} x \right)^2 = \frac{16}{9} x^2
\]

### Step 3: Set up the arc length integral
Using the formula for arc length, we now substitute into the integral:
\[
L = \int_0^3 \sqrt{1 + \frac{16}{9} x^2} \, dx
\]

### Step 4: Solve the integral
This is the integral we need to evaluate:
\[
L = \int_0^3 \sqrt{1 + \frac{16}{9} x^2} \, dx
\]

This integral can be solved using a trigonometric substitution or other advanced techniques. Let's compute the result of this integral numerically.

I will now calculate the value of this integral.

It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can still solve the integral numerically using a calculator or any mathematical software to find the exact arc length.

If you have any questions or need further clarifications, feel free to ask!

Here are 5 related questions:
1. How do you perform trigonometric substitution to solve integrals like this one?
2. What is the significance of arc length in geometry?
3. How does the complexity of the arc length formula change with different functions?
4. Can we use an approximation method for finding the arc length of complex curves?
5. What happens if we change the interval of integration for this problem?

**Tip:** Always remember to square the derivative before applying it in the arc length formula to avoid calculation mistakes.

y = 2/3 x^2 + 1/2. Find the arc length of the graph of the function over the indicated interval between 0 and 3.

This problem involves finding the arc length of the function y = 2/3 x^2 + 1/2 over the interval from 0 to 3 using calculus. Learn how to differentiate the function, square the derivative, and set up the integral for arc length. Suitable for high school and college students studying calculus.

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