Final answer:
To calculate the arc length of the curve x = y^2 from y = -1 to y = 2, use the integral S = ∫ √(1 + (dx/dy)^2) dy, where dx/dy is the derivative of x with respect to y, which is 2y for this curve.
Step-by-step explanation:
To find the formula for the arc length of the curve x = y^2 from y = -1 to y = 2, we can use the arc length formula for a function defined as x = f(y).
The arc length S of a curve between two points can be found using the integral:
S = ∫ √(1 + (dx/dy)^2) dy
For the curve x = y^2, we find the derivative dx/dy = 2y. Plugging this into the formula, we get:
S = ∫ from y = -1 to y = 2 √(1 + (2y)^2) dy
S = ∫ from y = -1 to y = 2 √(1 + 4y^2) dy
This integral can be evaluated to find the arc length. Remember, because we are dealing with a curve that is not a circular arc, statements about the relationship between arc length and radius of a circle do not apply directly to this situation.