Find the arc length of the curve over the interval(a) x = 13 (y2 + 2)3/2 from y = 0 to y = 1

To find the arc length of the curve over the interval, we can use the formula for arc length:

L = ∫[a,b] √(1 + (dy/dx)²) dx

In this case, we are given the equation x = 13(y² + 2)^(3/2) and we need to find the arc length from y = 0 to y = 1.

Step 1: Find dy/dx To find dy/dx, we need to differentiate the equation x = 13(y² + 2)^(3/2) with respect to y. Let's do that:

dx/dy = d/dy [13(y² + 2)^(3/2)] dx/dy = 13 * (3/2) * (y² + 2)^(1/2) * 2y dx/dy = 39y(y² + 2)^(1/2)

Step 2: Square dy/dx To simplify the expression inside the square root in the arc length formula, we need to square dy/dx:

(dy/dx)² = (39y(y² + 2)^(1/2))² (dy/dx)² = 1521y²(y² + 2)

Step 3: Set up the integral Now we can set up the integral using the arc length formula:

L = ∫[a,b] √(1 + (dy/dx)²) dx L = ∫[0,1] √(1 + 1521y²(y² + 2)) dx

Step 4: Evaluate the integral To evaluate the integral, we need to simplify the expression inside the square root and then integrate. This may require some algebraic manipulation and substitution.

Unfortunately, I cannot perform the integration here as it requires more advanced mathematical tools. However, you can use numerical methods or consult a calculus resource to evaluate the integral and find the arc length of the curve over the given interval.