Textbook Question
Equal integrals Without evaluating integrals, explain the following equalities. (Hint: Draw pictures.)
b. ∫²₀(25−(x²+1)²) dx = 2∫₁⁵ y√y−1 dy
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9. Graphical Applications of Integrals
Introduction to Volume & Disk Method
3:54 minutesProblem 6.5.12
Textbook Question
9–20. Arc length calculations Find the arc length of the following curves on the given interval.
y = 3 ln x− x²/24 on [1, 6]
Verified step by step guidance1
Recall the formula for the arc length of a curve \(y = f(x)\) on the interval \([a, b]\):
\[L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\]
Find the derivative \(\frac{dy}{dx}\) of the given function \(y = 3 \ln x - \frac{x^2}{24}\). Use the derivative rules for logarithmic and polynomial functions:
\[\frac{dy}{dx} = 3 \cdot \frac{1}{x} - \frac{2x}{24}\]
Simplify the derivative expression:
\[\frac{dy}{dx} = \frac{3}{x} - \frac{x}{12}\]
Substitute \(\frac{dy}{dx}\) into the arc length formula under the square root:
\[L = \int_1^6 \sqrt{1 + \left(\frac{3}{x} - \frac{x}{12}\right)^2} \, dx\]
Set up the integral for evaluation. At this point, you can either simplify the expression inside the square root further or use numerical methods or a calculator to approximate the integral value over the interval \([1, 6]\).
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Arc Length Formula
The arc length of a curve y = f(x) from x = a to x = b is found using the integral formula L = ∫_a^b √(1 + (dy/dx)²) dx. This formula sums the lengths of infinitesimal line segments along the curve, providing the total distance traveled along it.
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Arc Length of Parametric Curves
Derivative of the Function
To apply the arc length formula, you need the derivative dy/dx of the function y = 3 ln x − x²/24. The derivative measures the slope of the curve at each point, which is essential for calculating the integrand √(1 + (dy/dx)²).
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Definite Integration
After finding the integrand, you evaluate the definite integral from x = 1 to x = 6. This process sums the continuous values of √(1 + (dy/dx)²) over the interval, yielding the exact arc length of the curve between the specified bounds.
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