Find the length of the curve x=2y^3/2 from y=0 to y=3
  
*Round your answer to four decimal places.*


The length of a curve given by the function \( x = f(y) \) between two points \( y = a \) and \( y = b \) can be found using the formula for arc length:

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

In this case, the curve is \( x = 2y^{3/2} \), so we need to find \( \frac{dx}{dy} \):

\[
\frac{dx}{dy} = \frac{d}{dy}\left( 2y^{3/2} \right) = 3y^{1/2}
\]

Now, substitute \( \frac{dx}{dy} \) into the arc length formula:

\[
L = \int_0^3 \sqrt{1 + (3y^{1/2})^2} \, dy
\]

This simplifies to:

\[
L = \int_0^3 \sqrt{1 + 9y} \, dy
\]

Next, we compute this integral. I'll handle the integration and calculate the result.

The length of the curve \( x = 2y^{3/2} \) from \( y = 0 \) to \( y = 3 \) is approximately \( 10.9009 \) units.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How would the arc length formula change if \( x \) were given in terms of \( y \) differently?
2. Can we apply the same method if the curve is in polar coordinates?
3. What is the geometric meaning of the arc length formula?
4. How does arc length change if the curve is rotated about an axis?
5. How can you compute the arc length if \( y \) is expressed in terms of \( x \)?

**Tip:** When computing integrals involving square roots, simplifying the expression beforehand often makes the calculation easier.

Find the length of the curve x = 2y^(3/2) from y = 0 to y = 3. Round your answer to four decimal places.

Learn how to calculate the arc length of the curve x = 2y^(3/2) from y = 0 to y = 3. This problem explores calculus concepts like arc length, integration, and differentiation. Step-by-step solution suitable for advanced high school or college students.

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