91Ó°ÊÓ

Calculus integration is a fundamental tool used to find accumulated quantities, such as areas under curves and the lengths of curves themselves. The concept of integration involves adding up infinitesimally small quantities over a continuous range. This is especially useful in calculating the arc length of a curve, which can be thought of as a series of tiny straight line segments stretched along the curve.

When finding the arc length of a curve defined by a function like \(y = x^{2/3}\) from \(x = a\) to \(x = b\), a specific formula is used:

• \( L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \)
• This formula calculates the total length by considering both the horizontal and vertical changes along the curve.

This integration problem can be simplified by substituting a previously calculated derivative into the arc length formula, followed by simplifying the integrand before solving.
Understanding this concept requires practice in integrating different forms of functions, often needing techniques like substitution or pattern recognition to simplify the expression to a solvable form.

The power rule is one of the most straightforward differentiation techniques in calculus. It allows us to find the derivative of terms in the form of \(x^n\), where \ is any real number. The rule states that the derivative of \(x^n\) is given by \(nx^{n-1}\).

In our exercise, we applied the power rule to find the derivative of \(y = x^{2/3}\). Following the power rule:

• Derivative: \((2/3)x^{-1/3}\)
• The exponent \((2/3)\) drops down, and we subtract one from the original exponent resulting in \(-1/3\).

By finding the derivative using the power rule, we provide the necessary component for the arc length formula. This step requires precision as errors in differentiation will propagate, affecting subsequent calculations.
Differentiation is crucial here as it reflects the dynamic changes in the curve, which directly impact the arc length calculation.

Evaluating a definite integral involves calculating the accumulated value of a function between specified limits. It's a two-step process: first, find the antiderivative of the integrand, and then apply the limits to obtain a numerical result.

In our task, the integral is \(\int_{1}^{8} \frac{\sqrt{9x^{2/3} + 4}}{3x^{1/3}} \, dx\). To evaluate this integral:

• Simplify the integrand by substituting if necessary, which might involve finding a pattern or utilizing known identities.
• Calculate the antiderivative, ensuring all steps are performed correctly.
• Substitute the limits \((1, 8)\) into the antiderivative and take the difference to find the arc length.

This process often requires algebraic manipulation and a deep understanding of how substitution works in integration.
Ultimately, evaluating the definite integral provides a concrete answer to the problem, yielding the exact arc length of the curve from \(x = 1 \) to \ x = 8\.