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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.AAE.9
Chapter 8, Problem 8.AAE.9

Evaluate the limits in Exercise 9 and 10 by identifying them with definite integrals and evaluating the integrals.
lim (n → ∞) Σ (from k=1 to n) ln √(1 + k/n)

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1
Recognize that the given limit is a Riemann sum of the form \(\lim_{n \to \infty} \sum_{k=1}^n f\left(\frac{k}{n}\right) \Delta x\), where \(\Delta x = \frac{1}{n}\) and \(f(x) = \ln \sqrt{1 + x}\).
Rewrite the summand \(\ln \sqrt{1 + \frac{k}{n}}\) as \(\frac{1}{2} \ln \left(1 + \frac{k}{n}\right)\) to simplify the expression inside the sum.
Express the sum as \(\sum_{k=1}^n \frac{1}{2} \ln \left(1 + \frac{k}{n}\right) \cdot \frac{1}{n}\), which matches the form of a Riemann sum for the integral of \(\frac{1}{2} \ln(1 + x)\) over the interval \([0,1]\).
Set up the definite integral corresponding to the limit: \(\int_0^1 \frac{1}{2} \ln(1 + x) \, dx\).
Evaluate the integral \(\int_0^1 \frac{1}{2} \ln(1 + x) \, dx\) using integration techniques such as integration by parts to find the exact value representing the limit.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integral as a Limit of Riemann Sums

A definite integral can be interpreted as the limit of a sum of function values times small intervals, known as a Riemann sum. As the number of subintervals increases indefinitely, the sum approaches the exact area under the curve. Recognizing a limit of sums as a definite integral is key to evaluating such limits.
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Introduction to Riemann Sums

Properties of Logarithmic Functions

Logarithmic functions, such as ln(x), have properties that simplify expressions, like ln(√x) = (1/2)ln(x). Understanding these properties helps in rewriting the sum's terms into a more manageable form, facilitating the identification of the integrand in the limit process.
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Properties of Functions

Limit of a Sequence and Continuity

Evaluating limits involving sequences often requires understanding how functions behave as their inputs approach a point. Continuity ensures that the limit of the function values equals the function value at the limit point, allowing substitution inside the integral after identifying the integrand.
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Related Practice
Textbook Question

Finding volume

Let R be the "triangular" region in the first quadrant that is bounded above by the line y = 1, below by the curve y = ln x, and on the left by the line x = 1.

Find the volume of the solid generated by revolving R about

a. the x-axis.

Textbook Question

Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.

∫ dx / (1 + sin x + cos x)

Textbook Question

Finding volume

The infinite region bounded by the coordinate axes and the curve y = −ln x in the first quadrant is revolved about the x-axis to generate a solid. Find the volume of the solid.

1
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Textbook Question

Evaluate the integrals in Exercises 1–6.

∫ (arcsin x)² dx

Textbook Question

Finding volume

The region in the first quadrant enclosed by the coordinate axes, the curve y = e^x, and the line x = 1 is revolved about the y-axis to generate a solid. Find the volume of the solid.

Textbook Question

Use the substitution z = tan(θ/2) to evaluate the integrals in Exercises 41 and 42.

∫ csc θ dθ