Textbook Question
Consider the following curves on the given intervals.
b. Use a calculator or software to approximate the surface area.
y=cos x, for 0≤x≤π/2; about the x-axis
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9. Graphical Applications of Integrals
Introduction to Volume & Disk Method
8:14 minutesProblem 6.5.18
Textbook Question
9–20. Arc length calculations Find the arc length of the following curves on the given interval.
x = 2e^√2y + 1/16e^−√2y, for 0 ≤ y ≤ ln²/√2
Verified step by step guidance1
Identify the given function: \(x = 2e^{\sqrt{2}y} + \frac{1}{16}e^{-\sqrt{2}y}\), with the interval \(0 \leq y \leq \frac{(\ln 2)^2}{\sqrt{2}}\).
Recall the formula for the arc length of a curve defined as \(x = f(y)\) over an interval \([a, b]\):
\[L = \int_a^b \sqrt{1 + \left(\frac{dx}{dy}\right)^2} \, dy\]
Compute the derivative \(\frac{dx}{dy}\): Differentiate each term of \(x\) with respect to \(y\) using the chain rule, noting that \(\frac{d}{dy} e^{k y} = k e^{k y}\).
Square the derivative \(\left(\frac{dx}{dy}\right)^2\) and add 1 inside the square root to form the integrand:
\[\sqrt{1 + \left(\frac{dx}{dy}\right)^2}\]
Set up the definite integral for the arc length over the given interval and prepare to evaluate:
\[L = \int_0^{\frac{(\ln 2)^2}{\sqrt{2}}} \sqrt{1 + \left(\frac{dx}{dy}\right)^2} \, dy\]
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Video duration:
8mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Arc Length Formula for Parametric and Explicit Curves
The arc length of a curve defined by x = f(y) over an interval [a, b] is found using the integral L = ∫ from a to b √(1 + (dx/dy)²) dy. This formula measures the length of the curve by summing infinitesimal line segments along the curve.
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Arc Length of Parametric Curves
Differentiation of Exponential Functions
To find dx/dy for functions involving exponentials like e^(√2 y), apply the chain rule. The derivative of e^(g(y)) is e^(g(y)) * g'(y), where g(y) is the exponent function. Accurate differentiation is essential for setting up the arc length integral.
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Evaluating Definite Integrals with Logarithmic Limits
The interval involves limits with logarithmic expressions, such as y from 0 to (ln 2)/√2. Understanding how to handle these limits and simplify expressions involving logarithms is important for correctly evaluating the definite integral for arc length.
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