Find the lengths of the curves in Exercises 1-12 . If you...

Find the lengths of the curves in Exercises $1-12 .$ If you have a grapher,
you may want to graph these curves to see what they look like.
$$x=\left(y^{4} / 4\right)+1 /\left(8 y^{2}\right) \quad from \quad y=1 to y=2$$

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Key Concepts

Arc Length Formula

The arc length formula is a fundamental concept in calculus used to find the length of a curve defined by a function. For a curve expressed as x = f(y), the arc length from y = a to y = b is given by the integral of the square root of 1 plus the square of the derivative of the function with respect to y. This formula encapsulates the idea of summing up infinitesimal segments of the curve to obtain the total length.

Differentiation

Differentiation involves finding the derivative of a function, which represents the rate at which a quantity changes. In the context of finding a curve’s length, computing the derivative of the function is crucial because the derivative appears squared inside the integral in the arc length formula. Accurate differentiation ensures that the integrand correctly represents the instantaneous change in the function.

Definite Integration

Definite integration is a process in calculus used to accumulate quantities over a specific interval. When applying the arc length formula, definite integration is used to sum the contributions of every infinitesimal segment of the curve from one boundary point to another. It allows for the precise calculation of the total length of the curve over the specified interval.

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