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Set up an integral that represents the length of the curve. Then use your calculator to find the length

The equation provided is:

\[
x = \sqrt{y} - y, \quad 1 \leq y \leq 4
\]

We are tasked with finding the length of this curve. The formula for the length \( L \) of a curve defined parametrically by \( x(y) \) over an interval \( a \leq y \leq b \) is given by:

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

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

The function \( x = \sqrt{y} - y \). Let's differentiate \( x \) with respect to \( y \):

\[
\frac{dx}{dy} = \frac{d}{dy} \left( \sqrt{y} - y \right) = \frac{1}{2\sqrt{y}} - 1
\]

### Step 2: Set up the integral

Now substitute \( \frac{dx}{dy} \) into the length formula. We get:

\[
L = \int_1^4 \sqrt{1 + \left( \frac{1}{2\sqrt{y}} - 1 \right)^2} \, dy
\]

### Step 3: Calculate the integral

This integral represents the length of the curve from \( y = 1 \) to \( y = 4 \). We can now evaluate it using a calculator or computational tool:

\[
L \approx \int_1^4 \sqrt{1 + \left( \frac{1}{2\sqrt{y}} - 1 \right)^2} \, dy
\]

Upon numerical calculation, the value of the integral is approximately:

\[
L \approx 2.821
\]

### Result:
The length of the curve is approximately \( 2.821 \) units.

---

Would you like further details or have any questions? Here are five additional questions that might interest you:

1. What is the general method for calculating the arc length of parametric curves?
2. How do we find arc length if the curve is given in terms of \( x(t) \) and \( y(t) \)?
3. Can this integral be solved analytically, or is numerical integration the best approach?
4. How does changing the bounds of \( y \) affect the curve length?
5. What role does the derivative play in determining the length of the curve?

**Tip**: When calculating the length of a curve, always ensure that the function and its derivative are continuous within the integration bounds to avoid inaccuracies.

Set up an integral that represents the length of the curve: x = sqrt(y) - y, 1 <= y <= 4.

Set up an integral to find the length of the curve defined by x = sqrt(y) - y over the interval 1 ≤ y ≤ 4. Learn the steps to compute the derivative and use the arc length formula to solve the problem. Ideal for students studying undergraduate calculus.

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