91Ó°ÊÓ

To find the length of a curve using integration with respect to x, we use a specific formula. This formula applies when the curve can be expressed as a function of x, represented as \( y = f(x) \). The arc length \( L \) from \( x = a \) to \( x = b \) is given by the integral:
\[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]
In our exercise, we have the curve \( y = x^2 \) between \( x = 1 \) and \( x = 2 \). The derivative \( \frac{dy}{dx} \) of this function is \( 2x \). Substituting this into our formula, we get:

• Compute \( \left( \frac{dy}{dx} \right)^2 = (2x)^2 = 4x^2 \)
• Therefore, the expression under the square root becomes \( 1 + 4x^2 \)

Thus, the integral for the arc length with respect to x is:
\[ L_x = \int_{1}^{2} \sqrt{1 + 4x^2} \, dx \]
This integration gives us the total arc length of the curve segment defined by \( y = x^2 \) between the specified limits.

Similarly, we can also find the arc length by integrating with respect to y. This method is used when the curve can be expressed as \( x = g(y) \). The arc length \( L \) from \( y = c \) to \( y = d \) is given by:
\[ L = \int_{c}^{d} \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \, dy \]
In this problem, the curve is described by \( x = \sqrt{y} \) between \( y = 1 \) and \( y = 4 \). The derivative \( \frac{dx}{dy} \) is \( \frac{1}{2\sqrt{y}} \). So the steps are:

• Calculate \( \left( \frac{dx}{dy} \right)^2 = \left( \frac{1}{2\sqrt{y}} \right)^2 = \frac{1}{4y} \)
• The expression then becomes \( 1 + \frac{1}{4y} \)

Thus, the integral for arc length with respect to y is:
\[ L_y = \int_{1}^{4} \sqrt{1 + \frac{1}{4y}} \, dy \]
By evaluating this integral, we get the same arc length for the curve segment segment despite using a different approach.

The arc length formula is crucial for calculating the length of curves accurately using calculus. When determining arc length, it is essential to choose the right variable (either x or y) based on the given function's form. The general arc length formula for a curve given by \( y = f(x) \) or \( x = g(y) \) involves the square root of a sum of squares, representing the differential lengths in the horizontal or vertical directions. To verify that integrating with respect to x and y gives the same arc length, a common approach is to use substitution.

• In our exercise, substituting \( y = x^2 \) for the integral with respect to y confirms they reflect equivalent arc lengths.
• Both variables offer complementary views of the curve, ensuring that regardless of the variable used, the arc length measure is invariant.

This step confirms the robustness of the arc length formula and the consistency of calculus in providing a reliable tool for geometric measurements. This is particularly useful in fields ranging from engineering to physics, where precise measurements are vital.