Textbook Question
A torus (doughnut) A torus is formed when a circle of radius 2 centered at (3, 0) is revolved about the y-axis.
a. Use the shell method to write an integral for the volume of the torus.
9. Graphical Applications of Integrals
Introduction to Volume & Disk Method
2:17 minutesProblem 6.5.29a
Textbook Question
21–30. {Use of Tech} Arc length by calculator
a. Write and simplify the integral that gives the arc length of the following curves on the given interval.
y = 1/x, for 1 ≤ x ≤ 10
Verified step by step guidance1
Recall the formula for the arc length of a curve given by a function \( y = f(x) \) on the interval \( [a, b] \):
$$ L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx $$
Identify the function and interval: here, \( y = \frac{1}{x} \) and \( x \) ranges from 1 to 10.
Compute the derivative \( \frac{dy}{dx} \) of \( y = \frac{1}{x} \). Use the power rule or rewrite \( y = x^{-1} \) and differentiate.
Square the derivative \( \left(\frac{dy}{dx}\right)^2 \) and add 1 inside the square root to form the integrand:
$$ \sqrt{1 + \left(\frac{dy}{dx}\right)^2} $$
Write the integral for the arc length explicitly with the limits 1 to 10:
$$ L = \int_1^{10} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx $$
This integral can then be evaluated using a calculator or numerical methods.
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Key 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 L = ∫_a^b √(1 + (dy/dx)^2) dx. This formula calculates the length of the curve by summing infinitesimal line segments along the curve.
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Derivative of the Function
To apply the arc length formula, you need the derivative dy/dx of the function y = 1/x. The derivative measures the slope of the curve at each point and is essential for computing the integrand √(1 + (dy/dx)^2).
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Definite Integral Evaluation
After setting up the integral for arc length, evaluating it over the interval [1, 10] gives the total length. This may require simplification or numerical methods, such as using a calculator, especially when the integral cannot be expressed in elementary functions.
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