Textbook Question
Definite integrals from graphs The figure shows the areas of regions bounded by the graph of ƒ and the 𝓍-axis. Evaluate the following integrals.
∫₀ᶜ |ƒ(𝓍)| d𝓍
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9. Graphical Applications of Integrals
Area Between Curves
2:30 minutesProblem 7.1.74a
Textbook Question
ln x is unbounded Use the following argument to show that lim (x → ∞) ln x = ∞ and lim (x → 0⁺) ln x = −∞.
a. Make a sketch of the function f(x) = 1/x on the interval [1, 2]. Explain why the area of the region bounded by y = f(x) and the x-axis on [1, 2] is ln 2.
Verified step by step guidance1
Step 1: Recall that the natural logarithm function \( \ln x \) can be defined as the integral \( \ln x = \int_1^x \frac{1}{t} \, dt \). This means the value of \( \ln x \) represents the area under the curve \( y = \frac{1}{t} \) from \( t = 1 \) to \( t = x \).
Step 2: Sketch the function \( f(x) = \frac{1}{x} \) on the interval \([1, 2]\). The curve starts at \( f(1) = 1 \) and decreases to \( f(2) = \frac{1}{2} \). The area under this curve between \( x = 1 \) and \( x = 2 \), bounded by the x-axis, corresponds exactly to \( \int_1^2 \frac{1}{x} \, dx = \ln 2 \).
Step 3: Explain that since \( \ln 2 \) is the area under \( y = \frac{1}{x} \) from 1 to 2, this integral interpretation helps us understand the behavior of \( \ln x \) as \( x \) changes. The area grows without bound as \( x \to \infty \), showing \( \lim_{x \to \infty} \ln x = \infty \).
Step 4: Similarly, consider the limit as \( x \to 0^+ \). The integral \( \int_x^1 \frac{1}{t} \, dt = -\ln x \) represents the area under \( y = \frac{1}{t} \) from \( t = x \) to 1. As \( x \) approaches 0 from the right, this area grows without bound, implying \( \ln x \to -\infty \).
Step 5: Summarize that the integral definition of \( \ln x \) as the area under \( y = \frac{1}{x} \) provides a clear geometric interpretation of why \( \ln x \) is unbounded: it increases without limit as \( x \to \infty \) and decreases without limit as \( x \to 0^+ \).
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Natural Logarithm as an Integral
The natural logarithm function ln(x) can be defined as the integral of 1/t from 1 to x, i.e., ln(x) = ∫₁ˣ (1/t) dt. This integral representation connects the area under the curve y = 1/t to the value of ln(x), providing a geometric interpretation of the logarithm.
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Derivative of the Natural Logarithmic Function
Behavior of the Integral and Limits
As x approaches infinity, the integral ∫₁ˣ (1/t) dt grows without bound, showing that ln(x) → ∞. Conversely, as x approaches 0 from the right, the integral from 1 to x (interpreted properly with limits) tends to negative infinity, demonstrating ln(x) → −∞.
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Area Under a Curve and Definite Integrals
The area bounded by the curve y = 1/x, the x-axis, and vertical lines x = 1 and x = 2 is given by the definite integral ∫₁² (1/x) dx. This area equals ln(2), illustrating how definite integrals measure accumulated quantities and relate to function values.
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