Textbook Question
88. The region in Exercise 87 is revolved about the x-axis to generate a solid.
a. Find the volume of the solid.
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.4.61a
Evaluate ∫ x³ √(1 - x²) dx using:
a. Integration by parts.
Verified step by step guidance1
Identify the integral to solve: \(\int x^{3} \sqrt{1 - x^{2}} \, dx\).
For integration by parts, choose parts of the integrand as \(u\) and \(dv\). A good choice is to let \(u = x^{2}\) (since \(x^{3} = x \cdot x^{2}\)) and \(dv = x \sqrt{1 - x^{2}} \, dx\) to simplify the integral after differentiation and integration.
Compute \(du\) by differentiating \(u\): \(du = 2x \, dx\).
Find \(v\) by integrating \(dv\): \(v = \int x \sqrt{1 - x^{2}} \, dx\). To do this, use a substitution such as \(t = 1 - x^{2}\), then express \(v\) in terms of \(t\) and integrate.
Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\). Substitute the expressions for \(u\), \(v\), and \(du\) into this formula to rewrite the original integral in terms of simpler integrals to evaluate.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Parts
Integration by parts is a technique based on the product rule for differentiation. It transforms the integral of a product of functions into simpler integrals using the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely simplifies the integral, especially when one function becomes simpler upon differentiation.
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06:18
Integration by Parts for Definite Integrals
Substitution Method
Substitution involves changing variables to simplify an integral. By setting a part of the integrand as a new variable, the integral can be rewritten in a simpler form. This is particularly useful when the integrand contains composite functions, such as √(1 - x²), which suggests substituting u = 1 - x².
Recommended video:
07:33Euler's Method
Handling Powers and Roots in Integrals
Integrals involving powers and roots require careful algebraic manipulation. Expressing roots as fractional exponents and simplifying powers can make integration more straightforward. Recognizing how to rewrite expressions like √(1 - x²) as (1 - x²)^(1/2) helps in applying integration techniques effectively.
Recommended video:
05:58Intro to Power Series
Related Practice
Textbook Question
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 2 to 4 of 1/(s - 1)² ds
Textbook Question
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 0 to 2 of sin(x + 1) dx
Textbook Question
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from -2 to 0 of (x² - 1) dx
Textbook Question
Finding area
Find the area of the region enclosed by the curve y = x sin(x) and the x-axis (see the accompanying figure) for:
b. π ≤ x ≤ 2π.
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Textbook Question
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 0 to 3 of 1/√(x + 1) dx