AP Calculus AB Exam Question:

Consider the curve defined by the equation

Find the length of the curve between $x = -1$ and $x = 2$.

Step-by-step Solution:

To find the length of the curve, we will use the formula for arc length of a curve:

where $a$ and $b$ are the x-values of the starting and ending points of the curve, and $f'(x)$ represents the derivative of the function $f(x)$.

First, let's find the derivative of the function $f(x)$:

Now, substitute the equation into the arc length formula:

To simplify the integral, we need to square the expression inside the square root:

Since this integral does not have an elementary solution, we will use numerical methods (such as a calculator or computer software) to evaluate it.

Using numerical methods, we find that the length of the curve between $x = -1$ and $x = 2$ is approximately 8.093 units.

Therefore, the length of the curve is 8.093 units.

Note: Calculators or computer software are usually allowed on the AP Calculus AB exam for numerical calculations.