Textbook Question
14–25. {Use of Tech} Areas of regions Determine the area of the given region.
9. Graphical Applications of Integrals
Area Between Curves
1:28 minutesProblem 6.5.23a
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 = ln x, for 1≤x≤4
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 the interval: here, \( y = \ln x \) and the interval is \( 1 \leq x \leq 4 \).
Compute the derivative \( \frac{dy}{dx} \) of \( y = \ln x \):
$$ \frac{dy}{dx} = \frac{1}{x} $$
Substitute \( \frac{dy}{dx} \) into the arc length formula:
$$ L = \int_1^4 \sqrt{1 + \left(\frac{1}{x}\right)^2} \, dx = \int_1^4 \sqrt{1 + \frac{1}{x^2}} \, dx $$
Simplify the expression under the square root:
$$ \sqrt{1 + \frac{1}{x^2}} = \sqrt{\frac{x^2 + 1}{x^2}} = \frac{\sqrt{x^2 + 1}}{x} $$
So the integral becomes:
$$ L = \int_1^4 \frac{\sqrt{x^2 + 1}}{x} \, dx $$
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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 given by the integral ∫ from a to b of √(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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Arc Length of Parametric Curves
Derivative of the Natural Logarithm Function
For y = ln(x), the derivative dy/dx is 1/x. Understanding this derivative is essential to substitute into the arc length formula, as it determines the slope of the curve at each point.
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Simplifying the Integral Expression
After substituting dy/dx into the arc length formula, the integral often requires algebraic simplification before evaluation. Simplifying √(1 + (1/x)²) to √(1 + 1/x²) = √((x² + 1)/x²) = √(x² + 1)/x helps in setting up the integral for numerical or calculator-based evaluation.
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