91Ó°ÊÓ

The concept of a **definite integral** is key when computing the arc length of a curve. A definite integral, generally represented as \( \int_{a}^{b} f(x) \, dx \), computes the accumulation of quantities, such as area or, in this case, arc length, along a specified interval \([a, b]\).

• In the problem, we initially observe the curve described by the integral \( y = \int_{\pi/6}^{x} \sqrt{64 \sin^2 u \cos^4 u - 1} \, du \).
• This indicates the curve is defined by accumulating values from the function \( \sqrt{64 \sin^2 u \cos^4 u - 1} \).

The calculation of this integral helps us understand the length between two specified points on the curve. The magic of definite integrals lies in their ability to precisely calculate total values over an interval, providing the structure we need to solve complex problems related to curves and their properties.

Calculating the derivative is a crucial step in finding the arc length of a curve, as it plays a significant role in the formula for arc length. Differentiation involves finding the rate of change of a function, which, in this context, is vital.For the equation in part (a), we start with the integral representation of the curve:\[y = \int_{\pi/6}^{x} \sqrt{64 \sin^{2} u \cos^{4} u - 1} \, du\]Taking the derivative with respect to \(x\) using the Fundamental Theorem of Calculus, we get:\[y'(x) = \sqrt{64 \sin^2 x \cos^4 x - 1}\]

• This derivative tells us how the curve "moves" in terms of \(x\).
• Once we have \(y'(x)\), we insert it into the arc length formula, \( L = \int_{a}^{b} \sqrt{1 + (f'(x))^2} \, dx \), to find the arc length.

Understanding derivatives allows us to determine how much a curve is bending or changing, which is essential in measuring its length accurately.

Parametric equations are used to describe the coordinates of points that make up a geometric object, like a curve, with each parameter representing a point's position. In part (b), the curve is described by parametric equations:\[x = a \cos t + a t \sin t\]\[y = a \sin t - a t \cos t\]

• These equations express the curve in terms of a third variable \(t\), often representing time or a similar parameter.

For calculating arc length with parametric equations, we differentiate each of the parametric equations with respect to \(t\):

• \( \frac{dx}{dt} = a t \cos t \)
• \( \frac{dy}{dt} = a t \sin t \)

The arc length \(L\) is then found by integrating these derivatives:\[L = \int_{t_1}^{t_2} \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt\]This method highlights how parametric equations provide a powerful way to handle complex curves, especially when one variable alone can't express the curves sufficiently. Understanding and solving parametric equations enable precise calculations of various aspects, such as arc length, that are often not straightforward otherwise.