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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.PE.55
Chapter 8, Problem 8.PE.55

Evaluate the improper integrals in Exercises 53–62.
∫ from 0 to 2 of (1 / (y − 1)^(2/3)) dy

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Identify the type of integral: This is an improper integral because the integrand \( \frac{1}{(y - 1)^{2/3}} \) becomes undefined at \( y = 1 \), which lies within the interval of integration \([0, 2]\).
Split the integral at the point of discontinuity to handle the improper behavior separately: write \( \int_0^2 \frac{1}{(y - 1)^{2/3}} \, dy = \int_0^1 \frac{1}{(y - 1)^{2/3}} \, dy + \int_1^2 \frac{1}{(y - 1)^{2/3}} \, dy \).
Rewrite each integral as a limit approaching the point of discontinuity: for the first integral, use \( \lim_{t \to 1^-} \int_0^t \frac{1}{(y - 1)^{2/3}} \, dy \), and for the second integral, use \( \lim_{s \to 1^+} \int_s^2 \frac{1}{(y - 1)^{2/3}} \, dy \).
Evaluate the antiderivative of the integrand \( \frac{1}{(y - 1)^{2/3}} \). Use the power rule for integration by rewriting the integrand as \( (y - 1)^{-2/3} \) and then integrate: \( \int (y - 1)^{-2/3} \, dy = \frac{(y - 1)^{1/3}}{(1/3)} + C = 3 (y - 1)^{1/3} + C \).
Apply the limits of integration to the antiderivative for each integral and then take the respective limits \( t \to 1^- \) and \( s \to 1^+ \) to determine if the improper integrals converge or diverge.

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Key Concepts

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

Improper Integrals

Improper integrals involve integrals with infinite limits or integrands that become unbounded within the interval. To evaluate them, one must consider limits approaching the problematic points to determine if the integral converges or diverges.
Recommended video:
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Improper Integrals: Infinite Intervals

Handling Singularities in the Integrand

When the integrand has a singularity (point where it is undefined or infinite) within the integration interval, the integral is split at that point. Each part is evaluated as a limit approaching the singularity to check for convergence.
Recommended video:
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Completing the Square to Rewrite the Integrand

Integration of Power Functions

Integrating functions of the form (x - a)^p requires applying the power rule for integration, valid when p ≠ -1. For fractional exponents, careful attention is needed to handle the domain and convergence, especially near singularities.
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Representing Functions as Power Series
Related Practice
Textbook Question

A brief calculation shows that if 0 ≤ x ≤ 1, then the second derivative of

f(x) = √(1 + x⁴)

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Based on this, about how many subdivisions would you need to estimate the integral of f from 0 to 1

with an error no greater than 10⁻³ in absolute value using the Trapezoidal Rule?

Textbook Question

Evaluate the improper integrals in Exercises 53–62.

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